How do I install plumbing for a bathroom in a basement?

You must first locate the sewer main. A rough estimate can be obtained by noting where the stack comes down as per against where the sewer leaves the house. The sewer main should follow line of sight between the two. Dig a lateral trench to pinpoint the location. After you locate the main Most bathroom groups are roughed in like this.
Toilet connects to sewer main. Lavatory connects to toilet drain and runs a vent off the top the stubout tee out the roof or revents back into a dry vent in the attic.. The toilet wet vents through the lavatory vent and the tub/shower connects to the lavatory drain and is wet vented by it. This is a normal rough in and is acceptable both by local and state codes and also The Standard Plumbing Code Book in 90 percent of the country. Check your local codes.to make sure you're not in the excluded 10 percent.. The vent off the lavatory may be run out the roof or revented back into a dry vent in the attic or if you're reventing back into a fixtures dry vent you must make your connection at least 6 inches over that fixtures flood rim. Hope this helps, tom