Many, many institutions worldwide provide bachelors, masters and PhDs in geography. Is this required for GIS study or GIS work? Not necessarily. Is it a good grounding? I found my bachelors degree in Geography (from The University of Chicago) to be an effective overview and provided some insight into the kinds of problems GIS might be used to address. The most important Geography classes for GIS, in my opinion, include geographic analysis, statistics and cartography. The Association of American Geographers 1998 listing of programs in the U.S. and Canada is available on an interactive map. Some other websites covering worldwide offerings are listed here.
GIS degrees are relatively new. Bachelors degrees tend to be more broadly focused. A student is likely to receive a BA in say geography or another discipline, with a concentration in GIS. There's directory here. Masters and PhDs are more likely to be "in" GIS. A good GIS degree will also provide some of the basics of a geography degree. Most GIS degrees I've seen are at the masters and aim to create technically savvy users. These programs can also help students explore the use of GIS in a particular area such as forestry or business. Other programs focus on GIS technology and its impact in developing nations or culture. Few, if any Masters programs require a geography degree, but most require a bachelors that can somehow be "related" to GIS.
Product training options are exploding. From tutorials delivered with packaged software, to those provided by vendors and others that are available online (for example, here), to classroom and online teaching, there's something for everyone. "Do it yourself" methods tend to be relatively inexpensive and may be free online, or require just a book purchase. Many vendors now produce their own training books appropriate for use at home or in a classroom. ESRI's workbook series is quite outstanding.
Most vendors, and many universities teach classes aimed at introducing GIS and a specific software package. Vendors typically have the best directories of these classes. Most vendors will be happy to send instructors to you, at a price. Look for vendor or reseller led seminars at conferences to fill in gaps in specific areas. Sometimes these free or lowcost seminars can clarify quite a lot of technical information, though they may fall under the company's "marketing" budget.
And, see online training, below.
The fastest growing sector in education, it seems, is online training and education. Online bachelors and masters are popping up in nearly every field, including geography and GIS. One worldwide postgraduate program is UNIGIS.
Vendor classes are growing too. ESRI's Virtual Campus has free and fee-based classes and Intergraph has just begun offering online instruction. The important distinction in online training is whether it's self-paced (you work alone, at your pace) or instructor-led (you work, along with classmates and an instructor, within a specific timeframe, more like college). The latter tend to be more expensive but, from what I understand, more rewarding in the long run.
How to Find Education Options
If you are looking for a degree program, geography/GIS, start with academic geography organizations, like the AAG.
If you are looking at product training, start by contacting the software vendor's website.
Before investing in a program, track down people who've been through it to be sure it'll be worth your while. Newsgroups and conferences are great places to learn about others' experiences.