There is no train service to Tulsa, but you can always get there by Greyhound. The bus station is located right downtown at 317 S. Detroit Ave. Major routes head west to Las Vegas and Los Angeles, northeast to St. Louis and beyond, and south into Texas. Of course, you can get all over the U.S. with Greyhound's extensive network; it just might take a lot of time and patience.
With the dearth of speedy and reliable public transportation, not to mention the issues with taxicabs, driving is by far the easiest way to get around Tulsa. And one of the secret charms of Tulsa, one that not everyone realizes, is that it's laid out on a grid system, making it exceedingly easy to get around. Major streets are exactly one mile apart. North/south streets are named after U.S. cities (in alphabetical order, no less) and east/west streets are numbered. There are some quirks in the plan, but overall it makes for super-simple navigation.
Are you sure you wouldn't just rather rent a car? Tulsa is not known for its public transportation options. There's no metro, and the bus system, which is operated by Tulsa Transit, is considered a last resort for anyone who doesn't own a car and can't catch a ride. If you're staying downtown and aren't planning on doing much exploring, though, you can find routes to some key attractions. (If you're not staying downtown, don't even attempt it.)
Fares are $1.50 for adults and $1.25 for youth. Seniors over age 75 and children 4 and under ride free.
Bus #205 will get you to and from the airport and downtown.
With the exception of downtown, where most of the bus lines converge, the routes don't provide much coverage. The bus can get you into the general vicinity of where you're going, but routes can be miles apart.
Look for information on discount passes on Tulsa Transit's website. They offer 10-ride passes and also multi-day passes good for unlimited rides for one, seven or 31 days.