Here's an attempt of a quick (probably inexact) overview:
fry - frittieren / braten / backen.
roast - rösten / braten
This is one of the cases, where the distinctions are somewhat different (and pretty much arbitrary) in English and German. Also, it will depend on whether you're talking to a chef or someone who's never been in a kitchen, or someone in between.
There is a lot of overlap, though, so that in most cases braten will cover what you want to say.
Some examples where Germans use a different term than braten:
I'll try to add more examples :)
Oh my gosh there has to be some way of saying "fried eggs". In Yiddish we have praegeln, as in the following quatrain from the Ballad of Monisch by I. L. Peretz, in which the Devil and Lilith are having a quiet morning at home:
Sée shmuessen (chat) vertraut, vun dem un vun jenem;
Wemen zu packen ‘erein in Gehenum.
Wemen zu schmeissen mit eiserne rüter,
Un wemen zu präglen zu’m frühstuck mit putter.
I know nothing of this, but what I intended to be a comment has become long enough for an answer.
The words “rösten”, “braten” and “frittieren” all describe ways to prepare food that are relevant here. You can find their exact meaning by looking at the corresponding articles in the German language Wikipedia. There may be more words of interest to be found in the article “Garen”.
“Rösten” ist “roast”, but as in coffee beans. The pictures for “Braten” look very much like those for “roasting”, which should sufficiently explain this word. (The German page also links to the English page. This is a good way to use Wikipedia to find translations for things like these, very useful with animals and plants. Unfortunately the English page links to the page for “rösten”, which seems to be a bad choice.)
Finally, “frittieren” is definitely how you prepare French fries. That seems to be “deep-frying” in English.