You begin with a few different space vehicles.
The Delta Glider (DG) is a hypothetical single-stage-to-orbit (SSTO) vehicle capable of interplanetary flight. However, despite its fictitious powerplant and heat shield composition, it's still bound by a realistic physics system. Due to its small size and long range, though, it's far more forgiving. Expect to crash a few times, though. Though it takes off like an airplane from Earth, making that transition from atmospheric flight to orbit takes practice.
I reccomend running the "DG to the Moon" tutorial. This will give you an idea of what to expect, and it will also take you through some of the more esoteric and essential terms of space flight and orbital mechanics, things like apoapsis (the highest point on a given orbit, called "apogee" if the orbit is about earth, or "aphelion" when referring to orbits about the sun), or normal mode (an autopilot mode which points the spacecraft out of the plane of your orbit, used for adjusting the orbital plane), what a transition orbit is, how to do it, how to land on a moon when there's no atmosphere...and it'll involve you in the easier steps, also teaching you how to read the data displays and what the information means.
Space shuttle Atlantis. This is a realistic simulation of an actual launch vehicle. Bigger, heavier, and without the fuel capacity or engine power of the DG. Flying this takes patience, planning, and a lot more skill. The orbiter does not move as readily, so maneuvering is tricky. I haven't taken this one into orbit yet, I'm still learning the basics on the DG. However, I also reccomend running the "Atlantis to ISS" tutorial, as it'll demonstrate things like orbital planes and why they are important, and how to synchronize orbits.
Because it's a simulation, you can quicksave your state, or exit the simulation (Orbiter autosaves the last state of the simulator as "Current simulator state," so it picks up where you left off when you resume) where you can save your last simulation state in an independent file, or fast-forward time (the 3-day long flight to the moon can be sped up so you only spend an hour in realtime). Just remember to be in a stable orbit or trajectory with the autopilot turned off. The autopilot program can't calculate with time sped up 100 times or more, or you may find yourself slamming into something you were trying to avoid, or flying off into space. (If you're really careless you can end up killing yourself and your crew before liftoff by speeding up time with all the hatches closed and running out of oxygen.) The "time warp" feature is handy, just don't overdo it.
The program comes with documentation. If you can't find information on a particular vehicle in the main Orbiter manual, look for the relevant operations manual...it'll give you specific instructions on items specific to that spacecraft (SCRAM jets, hover thrusters, various keyboard commands). If all else fails, hold alt-shift, and it shows a list of keyboard commands on screen. Release alt-shift to clear your screen and return to the simulation.
Other ships include the dragonfly, a low-orbit shuttle used to move things around between orbits, Shuttle Alpha, a launch vehicle great for low-gravity worlds like the moon and mars, and any number of add-ons, including sci-fi starfighters and warp drive for those who just want to play around with "what if" scenarios.
Some things I found that'll make learning to fly to space easier include:
Go Play In Space by Bruce Irving. This is a great guide that puts terminology and techniques into plain English. Think of it like "Spaceflight for Dummies."
Dan's Orbiter Page is the home of "Orbiter Sound 3.5," the sound module for orbiter. This provides the user with engine sounds, hull noises, and audible warnings. He also hosts "Universal MMU," a module which allows you to perform EVA's in orbit.
Aerobrake MFD (Multifunction display) module is a great way to help plan re-entry paths, helping you predict when to burn engines to break orbit so that you can glide in over your destination spaceport.
And of course, there are plenty of really good add-ons that make an initial installation of orbiter complete...
Apollo Mission Simulator for Orbiter has complete meshes and documentation for flying the Apollo missions 8 through 17. Missions are even subdivided so you can skip ahead to different parts in a historical flight (say you want to take Neil's first steps on the moon...just skip ahead to the EVA), or play out the whole thing.
Altea Aerospace is the site which hosts the XR series of Delta Glider variants, from the upgraded Delta Glider XR1 (upgraded controls and an integrated cockpit system make this one easier to fly) to the XR5 Vanguard (the shuttle-scale SSTO vehicle).
You can even re-skin your vehicles. All you need to do is find the save (scenario) file, open it in a text editor, find your ship among the entries, and add a line:
SKIN [skin name here]
Here are some of the examples of the many available skins for download (all images drawn from their original source on Orbit Hangar Mods, another great source for lots of orbiter add-ons):
GAAAAHHH!!! GODS, WHAT HAVE I DONE?! MAKE IT STOP! MAKE IT STOP!!!
Anyway, it's a really cool game. And it's free.
In other news, I had a guest shot on "One Hour and a Half Later" by Mark Taylor.