Depends on the primer--some (actually, most now) are designed for either oil or water-based finish coats while others aren't. The cans you have will say.
You probably aren't restricted, see above. Premium paints from reputable vendors primarily. This is more a religious question than otherwise--everybody who posts in response will have their own personal favorite and horror story about somebody else's. In general, as in other areas, "you get what you pay for" is a valid saw.
Irrespective of age, surface prep is the most important factor and that has to happen prior to the prime coat going on. If you have old, weathered surfaces, cleaning and washing w/ a wood renewer containing oxalic acid is almost a necessity in order to get a fresh surface to which new paint will adhere. If there is existing tightly bonded paint, a good wash and scuff-sand will do the trick.