The answer is this:
Dodge is about equivalent to the combination of Unarmed and Gymnastics - if you full-dodge, you get the benefit vs. both melee and ranged attacks, whereas if you Gymnastic Dodge, you only get it versus one or the other. However, Gymnastics lets you specialize in Tumbling (ie, all your dodges), whereas Dodge requires range or melee. Likewise, you can pick something like "martial arts" for Unarmed that applies to your melee defense all the time.
Dodge is only one skill, thus, half as expensive.
Different things boost the skills. Gymnastics is boosted by Synthacardium cheaply; Dodge is boosted by Move By Wire as a side effect (ie, you would probably want MBW even if it did not add to Dodge). Other stuff that adds generally works on both.
Gymnastics comes "for free" with the Athletics group.
How I figure out what to take:
First, a lot of people dramatically overrate how good active defenses (ie, full defense) is in Shadowrun. It's not worthless, don't get me wrong, but it isn't really that good either. Offense is favored over defense, hitting first and hardest is favored over playing defensively - even if you full defense, you have excellent odds of getting hit anyways if you're fighting tough opponents. Spells and critter powers often bypass dodge entirely. Finally, full dodge is an awful option in team-on-team fights; smart enemies will stop shooting at you when you full dodge, and shoot at your buddies instead. Generally, investing in high Initiative, good attacks, and high passive defenses (Reaction, Reakt, Combat Sense, Body, Armor, and a melee defense skill) should be a higher priority for most people.
Second, do not half-ass active defenses. Being able to blow an action for a +1 to your defense is crap. Either invest in a decent amount of active defense, or don't bother at all.
If you already want either Athletics and Unarmed, don't take dodge. Just take those.
If you don't want either Athletics or Unarmed, just take Dodge.
Don't take Athletics unless you really care about either all three of Climbing, Running, and Swimming, or if you care about 2 of them and want Unarmed Combat anyways.
If you aren't sure, look at what 'ware you eventually plan on getting. If you plan on getting MBW, I would go dodge. If you plan on getting Synthacardium, go Gymnastics. Synthacardium is very cheap (30000 and .3 essence for +3 to athletics) so most people can fit it in easily at some point, but at the same time you may want MBW even without caring about the Dodge.
