While the most likely cause is the belts I would suggest checking the bearings first as it is very easy to do and the costs of having a bearing weld itself to the shaft spells quite an expensive repair, especially if it is a large framed alternator.
If you choose to do so then with the drive belt off rotate the alternator by hand and check for anything other than a dead smooth rotation. There should also be no axial (sideways movement) possible. The usual bearing to fail is the one at the pulley end and another useful check is after the alternator has been running a short time feel for any significant temperature difference of the alternator casing between the drive end and the rear end (watch your fingers).
If the belts seem tight enough already before you make any further adjustment to them then I would be inclined to suspect a bearing (or maybe other internal problem), not run the engine and take the alternator for a service
check as overtightening the belts will precipitate failure of any already sick bearing.
For the sake of good order
will repeat that I am not claiming it is the bearings, just that it is so simple to check those and their failure usually expensive that I am just suggesting that it would seem silly not to do so prior to anything else
If following Dan's advice you find you should replace the belt(s) then I would suggest that you go for a good brand, specifically Gates, in order to minimise further belt troubles and black dust throwing.