The first step to visiting Abkhazia is to obtain a clearance letter to enter it. Instructions are on the website of the Abkhazia Foreign Ministry: you fill in a form and email it to the foreign ministry's consular service along with a copy of your passport's personal data page. Within seven working days you should receive your clearance letter (in Russian) by email: check it carefully as mistakes are sometimes made.
Having entered Abkhazia with your clearance letter, you must then go within three working days to the consular service office, in central Sukhumi, to obtain your visa. Abkhazia visas are single pieces of paper, not stuck into your passport. A 10-day single-entry visa costs the rouble equivalent of US$11.
Georgia permits foreigners to enter Abkhazia by the Enguri Bridge boundary near Zugdidi, but treats entry to Abkhazia from Russia by the Psou River border as illegal entry into Georgia, punishable by a heavy fine or imprisonment under Georgian law. If you enter Abkhazia from Russia, don’t try to continue across Abkhazia’s southern boundary into undisputed Georgia. You'll need at least a double-entry Russian visa in advance, so that you can return to Russia.