Intro
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.
Checklist
Time
Usually less than two hours to connect new to old once pipes are run
Tools
Cutting and fitting tools for any type of pipe you will be working with, carpentry tools, reciprocating saw, hex-head driver
Skills
Cutting and joining the type of pipe used in the project
Prep
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.
Materials
Joining materials for the type of pipe you are using, transition fittings