Any student pilot can obtain any one of the three licenses, recreational, sport or private pilot without first obtaining a different license. Unfortunately, this also makes it more difficult to decide on which license is right for you.

The Recreational Pilots License
This is by far the most useless license one can obtain. On the hierarchy of license, it is on the bottom with the most restrictions. This license was designed for the farmer who had a runway on his field that he would use to spray his own crops.
Pros.................................................................Cons
Only requires 18hrs of instruction........................Will probably take at least 35 hrs anyways
........................................................................Can only carry one passenger
........................................................................Can't travel more than 50NM with additional training
..................................................................................(akin to the private pilot training)
.........................................................................Can't go into Class B, C, D airspace without additional training
..................................................................................(akin to the private pilot training)
.........................................................................Must carry logbook with those special endorsements
.........................................................................Only fly single engine airplanes certificated for four or less passengers
..................................................................................(180HP max without retractable landing gear)
.........................................................................Cannot fly above 10,000ft, with less than 3sm visibility or at night

There are a lot of limitations on the recreational pilots license. I think the FAA record shows that an average of 4 people get a recreational licese a year speaks a lot to its limitations.

Sports License This license was created to fill a niche group of pilots who wanted to fly but for some reason decided to let their FAA medicals. The main concern the FAA has with pilots without medicals is they don't want them killing other people so there is a limitation on the number of passengers. On the hierarchy of pilots licenses, this is above the recreational license but below the private pilots license. The biggest advantage is the ability to fly an airplane without a medical license.

Pros.................................................................Cons
Only license you can fly without a medical............Unless your medical application was denied
Only requires 20hrs of flight instruction.................Can only fly light sport aircraft
Fly anywhere you want in the U.S........................Can't fly above 10,000ft with visibility less and 3sm and at night
.........................................................................Can't fly into Class B, C, D airspace.

The recreational pilots license and the sports license are roughly the same with two major exceptions: the aircraft you can fly and the requirement for a medical.

Private Pilots License
This license will give the greatest amount of flexibility after obtaining your license but is also the most difficult to achieve requiring the most hours. Most student pilots are in training to receive a private pilots license. This license enables you to train for the commercial pilots license.

Pros.................................................................Cons
Most freedom once the license is obtained...........Requires 40hrs of flight instruction
Allows you to train and fly other aircraft category and classes of aircraft
Can climb up to 18,000msl without an instrument rating
Can carry as many passenger you want (within limitations of the POH)
Can fly above the clouds (provided you can descend in VMC conditions)
Can obtain an instrument rating that allows you to fly in the clouds

The most important thing to ask yourself is "What am I trying to do with my license after I obtain it?" If your plans will never be to fly in the clouds or become a commercial pilot than the recreational or sports license might be right for you. If you don't think you can pass a medical... maybe a sports license is right for you however please read FAR 61.53 stating that you can't fly if you have a known medical deficiency. Whatever you decide, you'll have to live the limitations on the license and you can always upgrade to the private pilot license if you choose.

One of the great things about the sports pilots regulations is that it allows any pilot (from ATP to private) to operate under the sport pilot rules whenever they desire. This means, that if you decide to get your private pilots license and later decide that going to the doctor is too much trouble, you can fly a light sport aircraft without a medical certificate under the sport pilot rule.