You most likely have iguanas tweaks for TC installed with Tconstruct. What you need to do is to make a flint pick and use it. You can upgrade each tool one time. so stone will upgrade to copper. Then o/c it is time to get a smeltery up and running. You can use clay buckets to get lava to start of.
The first thing you will notice is that the only tools you can make with wood are shovels, bows and battlesigns. You will need to hunt down some bone or flint to make pickaxe and axe heads. Bone and flint tools have been nerfed somewhat and are on a par with vanilla stone tools, only with slightly less durability. Wood as a material has also been nerfed dramatically.
To make getting flint from gravel less tedious, you can put 4 pieces of gravel next to each other in the crafting grid to get one flint. Gravel blocks no longer drop flint randomly. When digging gravel you will always get back gravel.
Until you smelt copper, you will not be able to make a sword. Don't panic as you can still make a flint axe and a bow to defend yourself. Once you have dug out some copper and aluminum with your flint tools and have access to a smeltery, you can upgrade your tool set and make your first sword.
Getting lava for the smeltery
You will need to craft a clay bucket (3 clay in the usual pattern) and fire (cook) it in a furnace to bring lava to the smeltery. When used to carry lava, the clay bucket is one use only and will be destroyed upon placing the lava.
A note on stone parts and tools
Stone can still be used to make all parts but those parts cannot be used to make tools. However, you will need them to pour aluminum brass over, to make the casts needed to mould metal parts. If you don't understand how to do this, refer to a Tinker's Construct guide on making metal tools - the process has not changed.
Changing parts of existing tools
Gone are the old days of making new tools with each new material and throwing away the old one. You can now change the parts of existing tools using the tool station or tool forge like you would add an upgrade (see screenshots for an example). You should do this every time because doing so, you will not lose any modifiers, xp or levels the tool has gained so far.
Tinker's construct tools now gain xp either hitting mobs or breaking blocks. The tool's current XP and chance to level up will be shown on it's tooltip (see screenshots for an example). Leveling up a tools gives benefits such as opening up new modifier slots, as well as giving random bonuses.
When you make your first copper pickaxe, you will notice that you cannot mine anything new with it. From this tier onwards picks can only mine upto the level of the material they are made of. In order to mine the next tier of material you will either need to harness the power of a mob head or boost the pickaxe with XP (shown as "Boost XP" in the tooltip).
Using a mob head
An alterative to leveling up your pickaxe is to use the power of a mob head to boost it's mining capability. Copper pickaxes need a skeleton or zombie head, so go kill a few mobs and get hold of one. The default chance for a head to drop is 5%, so on average you should get one every 20 mobs killed. Add the head as a modifier to the pickaxe and the mining level will be boosted (see screenshots for an example). You do not need a free modifier slot to add a mob head.
* The features described in the above few paragraphs can be disabled in the config if you do not like them. Disabling pickaxeBoostRequired means that tools will be able to mine the next tier material as soon as they are made *
The iron age
Once boosted, copper pickaxes can mine iron. Since you can now defend yourself propely you should be able to go deeper into the caves and find some iron, and upgrade your tool set once more. You will need to modifiy your iron pick with a creeper head or gain enough boost XP for it to gain a mining boost. Once you have done this you can mine tin and gold.
The bronze age
Back at your smeltery, three copper combined with one tin in the smeltery will mix to give you a bronze alloy. You will need a enderman head or more boost XP to access the potential of your bronze pick. With a bronze pickaxe, you can mine obsidian and diamonds for the first time. The next tool set you will want to make will either be obsidian or alumite (check the Tinker's Construct book in game for the alloy mix).
Into the Nether
Alumite and obsidian requires a wither skeleton skull modifier, after which you can mine the first nether metal - Ardite. Ardite tools - which require a nether star (or more boost XP) to boost mining level - are needed to mine the other nether ore, cobalt. Finally Ardite and Cobalt can be combined to make the top tier tool material - Manyullyn.