。
She'll get along well with my grandmother.
Literal
She [topic-は] my grandmother [with-と] friendlily will-get-on.
仲良くやっていく is the standard collocation for 'get along well with someone, make it work harmoniously.' 仲良く ('in a friendly way, on good terms') is the adverbial form of 仲良し ('close, on good terms'). やっていく (literally 'go doing') is a compound with idiomatic 'get on, manage, make a life' meaning — used for both relationships and survival. と marks the partner with whom one is getting along. The ~でしょう ending hedges the prediction.