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(Updated answer, added some more details and updated information on the new Flamethrower)

Early on Shotguns and Poison capsules are some of the best as noted by other answers. Poison capsules will slowly kill any trees in their AOE, while Shotguns spread allows easy destruction by shooting them, and can be upgraded to shoot faster.

Flamethrowers are also good, but updates have changed how good they are. Previously, they were like shotguns, better early, but worse later since shotguns could be upgraded, but now they have their own mechanics which make them better, but at a cost. Now you can use very little fuel and light the forest on fire, which over time removes (need a fact check here, cant find any solid numbers) 90% or so of the trees. The fire spreads relatively slowly, and will only jump a short distance to nearby trees, but a couple of fuel will eventually light most of the forest on fire. Its slower than the combat shotgun with a few speed upgrades, but costs way less ammo, though it also releases a lot of pollution.

Later on in the game, you can have drones Harvest them for you by highlighting them with the deconstruction plans. Though this is limited to the Green and Orange zones around Roboports. You do need Logistics storage chests (yellow chests that require a special research) for the bots to place the wood in. They must be in the same Logistics Network as the Roboport which is supplying the construction zone being used to remove the trees to be used as a drop off point.

Also be careful when the Roboports construction zone overlaps your personal one when building or deconstructing. Personal Roboports will be assigned first for most tasks, since you are the closest, but if more construction bots are needed than you can supply in that instant, it will use the Logistics networks bots to fill the gap, rather than wait for your personal bots to complete their first actions. If you only have one personal Roboport, and need to deconstruct 15 trees, 10 bots will come from your personal inventory, but if 5 trees are not assigned bots from your personal Roboport, but are in the construction range of your logistics network, then 5 construction bots in that network will be assigned to it. If there are no bots in the network, then the trees are unassigned, and when your personal bots finish the first trees, they will be assigned to it.

One of if not the best very early game solution is Grenades. They are cheep, costing only coal and iron plates, and with one damage upgrade (as of .13) they can 2 shot trees in a small-medium AOE. Very early game, they can destroy forests quickly and cheaply, and even into late game can be a solid choice over some of the other solutions.

Shotguns and Poison capsules are also decent. Poison Capsules take a little longer to get to, but have a large AOE which will kill just about any trees inside, simply coat the trees and wait a few seconds. Shotguns are a bit slower initially, but with some upgrades can be a good solution as well. A Combat shotgun with upgrades will decimate trees.

Flamethrowers used to be like shotguns, but updates have made them far better. For a few fuel, you can light some trees in the middle of the forest on fire and just wait. Fire spreads to nearby trees, so even a small fire will burn an entire dense forest. Afterwards, about 10% of the trees will remain as burned trunks, and you can shoot them with upgraded shotguns for fast cleanup. This has the advantage that you can start the fire and go do other things for a few minutes, but it will release a lot of pollution, and damage any of your buildings if they are too close to the trees on fire.

Later on in the game, you can have drones Harvest them for you by highlighting them with the deconstruction plans, though this is limited to the Green and Orange zones around Roboports. You do need Logistics storage chests (yellow chests that require a special research) for the bots to place the wood in. They must be in the same Logistics Network as the Roboport which is supplying the construction zone being used to remove the trees to be used as a drop off point.

If you do have both Roboports and Personal Roboports overlapping, then you need to be careful. If you highlight 100 trees, but your Personal ports only allow 50 drones at once, then your Logistics network will supply the other 50 if it can rather than waiting for your personal ones to finish. The only way around that to not have any construction bots in that specific Logistics network, so that it can't do anything, and the other 50 trees remain unassigned a bot until your personal ones become available again.

Early on, Flamethrowers, Shotguns, and Poison capsulse are the best as noted by other answers.

Later on in the game, you can have drones Harvest them for you by highlighting them with the deconstruction plans. This is limited by where you place Roboports, so using Long range power poles to place one down in the middle of the forest with a storage chest would work. Another option you can also use is Modular armor and the Personal roboports to do so on the go. A personal roboport allows for 10 bots in a moderate area around you to construct and deconstruct things, with 2 charge ports for them before they go back into your inventory. Adding additional ports to the armor increases the range, the number of bots by 10, and adds 2 more recharging ports. Your personal AOE for building and deconstruction can get significantly bigger than a regular roboport, and is always centered on you allowing you to take down entire forests a few minutes while barely moving. Note that you will need the inventory space for the wood if you use the personal roboports, and probably also a portable fusion reactor to power everything.

(Updated answer, added some more details and updated information on the new Flamethrower)

Early on Shotguns and Poison capsules are some of the best as noted by other answers. Poison capsules will slowly kill any trees in their AOE, while Shotguns spread allows easy destruction by shooting them, and can be upgraded to shoot faster.

Flamethrowers are also good, but updates have changed how good they are. Previously, they were like shotguns, better early, but worse later since shotguns could be upgraded, but now they have their own mechanics which make them better, but at a cost. Now you can use very little fuel and light the forest on fire, which over time removes (need a fact check here, cant find any solid numbers) 90% or so of the trees. The fire spreads relatively slowly, and will only jump a short distance to nearby trees, but a couple of fuel will eventually light most of the forest on fire. Its slower than the combat shotgun with a few speed upgrades, but costs way less ammo, though it also releases a lot of pollution.

Later on in the game, you can have drones Harvest them for you by highlighting them with the deconstruction plans. Though this is limited to the Green and Orange zones around Roboports. You do need Logistics storage chests (yellow chests that require a special research) for the bots to place the wood in. They must be in the same Logistics Network as the Roboport which is supplying the construction zone being used to remove the trees to be used as a drop off point.

The second Roboport Option is the Personal Roboports. If you have Moduarly added Personal Roboports to your mid/end game armor, then you can take the construction zone with you, and Construction bots in your inventory will build and deconstruct stuff for you. They will deconstruct trees and place the wood in your inventory, so long as it is within your own personal construction zone. Each additional Personal Roboport modded into your armor increases the size of the construction zone that centers around your character, and allows 10 more drones to work. Note that you will need a lot of power for these to work, so for any serious amount of speed, you will need portable fusion reactors to power them.

Also be careful when the Roboports construction zone overlaps your personal one when building or deconstructing. Personal Roboports will be assigned first for most tasks, since you are the closest, but if more construction bots are needed than you can supply in that instant, it will use the Logistics networks bots to fill the gap, rather than wait for your personal bots to complete their first actions. If you only have one personal Roboport, and need to deconstruct 15 trees, 10 bots will come from your personal inventory, but if 5 trees are not assigned bots from your personal Roboport, but are in the construction range of your logistics network, then 5 construction bots in that network will be assigned to it. If there are no bots in the network, then the trees are unassigned, and when your personal bots finish the first trees, they will be assigned to it.

Early on, Flamethrowers, Shotguns, and Poison capsulse are the best as noted by other answers.

Later on in the game, you can have drones Harvest them for you. This is limited by where you place Roboports, but you can also use Modular armor and the Personal roboports to do so on the go. a personal roboport allows for 10 bots in a moderate area around you to construct and deconstruct things, with 2 charge ports for them before they go back into your inventory. Adding additional ports to the armor increases the range, the number of bots by 10, and adds 2 more recharging ports. Note that you will need the inventory space for the wood.

Early on, Flamethrowers, Shotguns, and Poison capsulse are the best as noted by other answers.

Later on in the game, you can have drones Harvest them for you by highlighting them with the deconstruction plans. This is limited by where you place Roboports, so using Long range power poles to place one down in the middle of the forest with a storage chest would work. Another option you can also use is Modular armor and the Personal roboports to do so on the go. A personal roboport allows for 10 bots in a moderate area around you to construct and deconstruct things, with 2 charge ports for them before they go back into your inventory. Adding additional ports to the armor increases the range, the number of bots by 10, and adds 2 more recharging ports. Your personal AOE for building and deconstruction can get significantly bigger than a regular roboport, and is always centered on you allowing you to take down entire forests a few minutes while barely moving. Note that you will need the inventory space for the wood if you use the personal roboports, and probably also a portable fusion reactor to power everything.