My basement was not stubbed. I want to put in a full bathroom with sink, shower, toilet and proper venting. I know the shower and the sink will need p-trap and each will need to have vent, since they are each more than 5 feet from main and each other. I am not sure that the toilet needs a p-trap, since the toilet has a trap built in. Is this correct? If I do install a p-trap for the toilet, will it still drain correctly, or will it create a back pressure between the toilet trap and the underground trap? I do not want to mess this up and have major re-work later! Thanks.

You do not install traps in the line for a toilet. The trap is built into the toilet itself. Everything else gets a trap regardless of distance. It is the venting that must be with in specified distances. Tom will give you the diagram and distances I'm sure.

Correct, we have several plumbers here and maybe you would like to submit your plumbing design so they can take a peek. They can help you determine pipe sizing and slope since you probably have little experience with plumbing.

Them experts are correct. Toilet bowls have the traps built in.
We rough in our bathrooms like this. (see image) More questions? I'm as close as a click. Tom