The two banks most expats use are TBC and Bank of Georgia. Both run English-speaking branches in Tbilisi and Batumi, and both offer multi-currency accounts — GEL, USD and EUR on one card.
What to bring: your passport. Some branches ask for proof of address or a reference; requirements vary by branch and by nationality, and they change, so call the branch first rather than trusting a forum post.
Cards work almost everywhere in the cities. Cash still matters in markets, in taxis outside the apps, and in the mountains.
This page is re-checked because bank onboarding rules for foreigners have changed more than once.