"Gå" and "åker" have similar, though different meanings. Though both are often translated into English as "to go," the verbs imply two different means of going. "Gå" basically means to travel somewhere on foot. (There are instances in which it means other things, though these usually don't apply to humans. For example, "Går det har busen till Stockholm?" [Does this bus go to Sweden?]) Åka is really traveling somewhere with the help of some sort of motor vehicle or other device - jag åker bil [I go by car], jag åker cykel (I go by bike), etc. This gives a pretty good explanation: http://blogs.transparent.com/swedish/aka-or-ga/. To sum it up: if you're going by foot, gå. If you're getting from one place to another on something: åka.
It depends. We move forward can be used in several different senses. If you're using it in an abstract sense, går would generally be the best choice. If you're referring to physical movement, like 'the train was stuck for several hours but now we're moving forward', we'd say rör oss.
Edit, I reread your question and I see I answered it a bit backwards. Still what I wrote was probably useful too so I'll leave it. åker framåt wouldn't normally be used to mean 'move forward', but it could mean that in the sense of 'are slipping/sliding forward'. Like when you slide forwards in your seat for instance.
The other way around – åker is only for going by some kind of vehicle.
går is basically only for walking on foot, although as we've discussed on this page, vehicles themselves can also gå – Bussen går till Stockholm 'The bus goes to Stockholm'. But you can't use gå for a person who goes by bus.