If you want to make a mining cart with the cart assembler, you'll need an engine (obviously), a drill of some kind, plenty of storage, a railer, a bridge builder, and a torch placer unless you don't mind lots of mobs spawning in your tunnels. You could also add the cleaning module, or whatever it's called, to clean up anything that gets dropped on the ground (I know Xycraft crystals tend to get dropped when the blocks they're attached to are destroyed).
If you just need more resources in general, rather than looking for anything specific, you're probably best off with a quarry if you have the resources to both build and power it. If you don't have the resources for that yet, I'd suggest manual mining until you do. Branch mine if you're looking for specific resources.
Tunnel bores and mining carts and such are great, but if you're looking for more than one or two specific resources, you'll end up missing out on some stuff since they mine horizontally rather than vertically, so you can't, for example, find diamonds, copper, AND tin with the same tunnel bore unless it's digging out a really tall tunnel.
EDIT: As for turtles, which I totally forgot to mention, they're easy to use for simple things like digging a tunnel. As a matter of fact, you can just open up a turtle, put some fuel in the first slot, then at the prompt you type "tunnel X" where X is the length of the tunnel you want. It'll dig a tunnel as long as you told it to, three blocks wide and two tall. Great for semi-automated branch mining. It'll stop if it runs out of fuel or inventory space though.
So "tunnel 20" would dig out a 3x2x20 tunnel. It's not the fastest method, but it's pretty easy.