My base is in a pretty forested area and I have to cut down trees before I build pretty much everything. Is there something I can research or build or whatever that will allow me to get rid of a lot of trees in one go? I don't care about retrieving wood from them.
You can use your weapons to clear them out. Early on, shotgun fire and grenades are significantly faster than cutting them down by hand.
In addition to using weapons there is also a possibility that is less work intensive and provides you with usable wood:
Use Construction robots and the deconstruction planner (red blueprint). This does however require technologies which become available quite late in the game, so Jay Lapham's answer maybe more applicable in many cases.
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.
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.
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.
The fastest and easiest way to clear trees is with a Poison Capsule, which I typically set up for manufacture. They'll clear trees in a very large radius; toss and wait. Very little player interaction involved. The recipe comes from Military 3, whose cost includes 50 blue potions.
Grenades and shotguns will also clear trees; they are more time-consuming to kill trees but require much less tech.