I had to talk to a lot of people to actually figure out the language situation. Ethnologue claims that there are over 60 languages in this small country, but I think that number is more a product of bored linguists splitting hairs. I don't know much about the northern half of Ghana, but here is how the southern half breaks down:
Twi - This is the lingua-franca of Ghana. Just about anywhere you go from Accra to Kumasi to Takoradi, people will understand Twi. It's not too difficult to pick up: Twi speakers have generally dropped much of the tonality, due to the influx of second-language speakers.
Fante - This is the high-falutin version of Twi spoken in the Cape Coast vicinity. Think Twi with tones. Fante speakers understand Twi fine, but they have their own way of pronouncing everything. If you try to speak Fante to them, they will smile and laugh at your pathetic attempts. The consonants are very complicated.
Ga - This language is spoken in parts of Accra and east to Tema. Just like Twi, it is Dangbe stripped of tones.
Dangbe - Spoken in the south-eastern part of the country, up into the lower Volta Region. It seems much easier to pick up than Fante--I haven't noticed too many places where tones are crucial, but my knowledge of the language is limited.
Krobo - Spoken in Koforidua. I haven't noticed much difference between this and Dangbe, and they seem to tolerate my Dangbe phrases just fine.
Ewe: Is its own language, and I only know a tiny bit of it. They use it in the Volta Region and in Togo. It sounds kinda Frenchy to me.
English: Is supposedly the official language of Ghana, but if you want to conduct more than the basic touristy conversations with the rank-and-file Ghanaian, you will need either an interpreter or some language training. English is much more common in the cities, so you might get by there, but English seems rare in the villages. Don't be too concerned: in general, most things that an obruni would be interested in, the Ghanaians have figured out the English required to make the transaction work.
Brong: If for some reason you travel west of Kumasi, you will need Brong. I'm not sure if it's close to Twi.
North of Kumasi you start getting into completely different stuff: Wolof, Frafra, Dagaare, some Hausa, and who knows what else. Most travelers to Ghana don't go north of Kumasi, but if for some reason you do, you'll need to do a little digging.
This is what I've learned so far. There's a different language in every village, but they're generally a dialect of either Twi (west of Accra) or Ga (east of Accra).