Most towns have dive shops, just check with them to see their course schedules - they usually offer either in the evening or on the weekends. You do need to pass some tests - swimming test, and the theory tests. All PADI diving centers can do referral check out dives - once you successfully pass the open water check out dives, you are certified for the beginner level.
The usual depth for Open Water level is 60-70ft, but you can do 80ft with a local, experienced divemaster without much problem. Of course you will do Safety Stop on ascend - it can be a floating stop or a horizontal stop (swim at that level for the required time).
Diving packages usually are cheaper than booking hotel and diving separately. Cruiseships these days can also be a great place to go for diving excursion. PADI has a program with Princess cruiseline, it is called New Wave. You can get a DVD or Video for the course ahead of time and then spend 2 mornings in classroom when you first board (usually they hold these on a Sea Day) and the afternoons are spent in the pool learning the actual skills. Then you will join the diving excursions and do your open water check out dives. This type of programs are getting quite popular now, particularly for people that only one member of the couple is learning and the other member can enjoy other things on cruiseship so both are happy.