I take it you don't have much experience with this stuff, hence the question about DIY stuff.
For the drain line, you can buy a cap out of PVC (white) or ABS (black) if its plastic pipe, and the right cement/solvent for that material. Then simply cut off the pipe where you want to, nice and straight, clean up the edges a bit if they're rough, and glue on the cap. You could also use a Fernco cap, which is a rubber cap with a steel band around it that you tighten. If the plumbing is metal, you'll have to go this way, and cutting the pipe off will require a reciprocating saw or angle grinder with a good blade/cutoff wheel.
For the copper, you'll have to turn off the main water supply for the house, and then drain the lines (open the lowest faucet in the house to drain some water out of the system, then use a bucket to open up the supply valves that you're cutting out, and drain as much out there as possible). Closing off the copper (I'm assuming you have copper water lines?) is a bit trickier if you don't have any experience with soldering. You can, but I strongly suggest against it, just buy a Sharkbite cap for each line and press them on. I think these are garbage and you're asking for trouble though. You might want to look into the presoldered fittings, you just put flux on the line, slide the fitting on, and then heat it up. It takes a little of the guesswork out of the equation.
Honestly though, if you have no experience with this stuff, get a handy friend, or pay a handyman for an hour of his time to come do it for you. It may cost less than the tools/parts you'd need to do it yourself.