GAMESS crash course?
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While I do think you should learn some theory, it's definitely possible to learn some practical computational chemistry through experimentation. And yes, there are a lot of abbreviations to follow.

One thing you should track while you explore is how long different calculations take. In general, more accurate quantum chemical methods are also more computationally costly. This means that they take longer, but also as systems get bigger, the cost rises faster. You might hear some people talk about "linear scaling" (i.e., double the size of the system, twice as much time) or $N^2$, $N^3$, etc.

A good resource, IMHO, is Molecular Modeling Basics by Jensen. It's available through CRC Press but Jan also blogs extensively. He uses GAMESS and Avogadro but also Jmol and other programs.

While the Avogadro project is working on some more detailed tutorials, I'd suggest starting some calculations with HF/6-31G* or B3LYP/6-31G* and see how things go for different molecules. I strongly suggest running some calculations and then consider how the results compare to experiment. Benchmarking is important. Computational models all have systematic and other errors, and for real work it's critical to run a set of molecules and carefully choose methods that work well.

Here are a few key terms to consider:


Method: There are several types of methods.


Semiempirical methods: These include AM1, PM3, RM1, (and PM6, PM7, and others that I don't think are in GAMESS-US). Such methods are fast, since they approximate some integrals and use empirical parameters. They also have an implied basis set which is usually very small. (Avogadro doesn't support viewing semiempirical calculations with GAMESS yet.)
Wavefunction methods: These include HF and relatives. These are similar to typical quantum chemistry methods taught in school.


MP2, MP3, MP4: These are perturbation methods that improve on Hartree Fock and introduce some level of electron correlation.
CCSD, CCSD(T), etc.: These are "coupled cluster" methods that are highly accurate, but often slow, and do not handle large molecules.

Density Functional: This includes another large set of methods, including BLYP, B3LYP, and newer methods like M06-X. Rather than approach things through the Schrödinger equation, 

Basis: Most methods (except AM1, RM1, PM3, etc.) also require a "basis set," a set of functions to describe the atomic orbitals. Wikipedia article There are a few families, including:


Pople Type: 3-21G(d), 6-31G(d), 6-311+G(d), etc. These are older, more traditional basis sets. Generally 6-31G(d) or 6-31G* is considered minimally acceptable today. The "*" or (p,d,f) included at the end indicate that higher polarization is included on some or all atoms. 
Correlation Consistent: cc-pVDZ, aug-cc-pVTZ, etc. These are newer basis sets, designed to function better with methods that treat electron correlation. The "aug-" includes various diffuse functions, which are useful particularly for anions and electron density that is particularly, well, diffuse. (In Pople basis sets, these are indicated with "+".