Battery intermittently flat, also fan belt breakage

Two problems

1)Fan belt problems over the past year: squeaky when starting, increasingly bad over time, has been tightened and checked at garage. First time replaced because it was virtually shredded. Three weeks later it broke again while driving (resulting in no brakes & steering!). Each time it's replaced/tightened it's fine for a few weeks then starts squeaking again.

2) Intermittent flat battery. Car starts but if only a short journey then car wont start again - sometimes so flat even a jump start wont sort it. Doesn't seem to give any warning signs.
Battery only couple of years old, garage checked it, fine. Alternator checked, charging ok. Everything checked again at garage only last week, but still having to put battery on overnight charge at least once a week.

Any ideas? (in plain English please as I'm not really a car person).