The most common way to connect a new drain, vent, or supply line to an existing line is to install a tee fitting.
Shut off water to existing supply pipes and drain the lines. Flush all toilets and caution others not to use drains. After opening a drain line, make sure no one uses a sink or faucet that drains into it. Seal any open drain lines with a rag to block fumes.
If joining pipes of different materials, make sure the transition fitting conforms to local code.
Typically it doesn't matter exactly where you join to an existing pipe, but the new service must be precisely located. So it's usually easier to start pipe runs at the new location and travel toward the existing pipes rather than vice versa.
Usually less than two hours to connect new to old once pipes are run
Cutting and fitting tools for any type of pipe you will be working with, carpentry tools, reciprocating saw, hex-head driver
Cutting and joining the type of pipe used in the project
Run new pipes from the new service location to the existing pipe; install the last pipe a little longer than it needs to be so you can cut it to length when you make the connection to the old pipe.
Joining materials for the type of pipe you are using, transition fittings
To join a new plastic drainpipe to an old steel pipe, run new pipe into the room. Prime and glue two 5-inch pieces of pipe to a tee fitting. Temporarily run pipe -- longer than it needs to be -- so it comes near the old pipe. (When running pipe across a stud wall, you may need to notch-cut some of the holes, using a reciprocating saw.) Dry-fit the tee assembly onto the new pipe and hold it next to the existing pipe. Mark the existing pipe for cutting. You may need to cut the opening larger than the tee assembly to accommodate the neoprene sleeves on the banded couplings.
To tap into existing copper lines, shut off the water. With a tubing cutter cut an opening in each pipe that is about an inch shorter than a tee fitting. Dry-fit the tees. If the pipes are rigidly installed, remove a clamp or two so you can pull the pieces apart slightly.