Tomesh,
There are many climbs in the world that knowing glacier travel and basic rope work will be sufficient to travel safely. I have friends that have climbed a lot in the Himalayas on non-guided trips who are very competent in what some people would classify as mountaineering but are mediocre rock climbers and ice climbers. Any additional skill sets you develop is only beneficial, be it technical rock climbing and ice climbing. You have to decide what you would like to do in the future.
Now learning at the climbing gym, where nowadays many people learn, will teach you only a few things. Belaying being one. Face climbing is another, and some gyms have cracks. Not much else can be learned at a gym. Its also unfortunate, the quality of learning experience varies tremendously from gym to gym. In Flagstaff, the staff tend to be very active first ascentionists locally and are good resources. Some gyms in the Bay Area and in New York (where I haved lived before) tend to be operated by people who have a mediocre or no knowledge of climbing I hate to say (not all be some that I have seen).
Learning at one of the meccas of climbing of the style you'd like to learn (in the US and Canada that would be Yosemite, Squamish, Indian Creek, Gunks, for rock, Canadian Rockies, the Tetons, Sierras for Alpine climbing, or the Cascades for mountaineering), you will have a better chance to have guides that are extremely experienced and active in that medium, as long as they are also effective teachers.