There was a time when I would argue for an oil-based stain solid color stain in this situation but not necessarily anymore. I would think about latex acrylic and something like Sherwin Williams Woodscapes solid color stain. It will even out any color anomolies and you can get it mixed to match any color. It is great stuff. MAB made a great solid stain as well but since it was acquired by SW I assume the products are the same. I've never used the Benjamin Moore product even though I used their paint most. Solid color stains do not require primer.
If you are overly concerned about the dryness of the wood you could apply a mix of 1:1 processed, not raw, lindseed oil and mineral spirits, paint thinner or similar solvent as a sealer. Wait for the solvents to evaporate then stain or prime and paint.
If you want to paint it, I would use Benjamin Moore Fresh Start alkyd primer (solvent based) and follow with two finish coats of quality acrylic house paint.
I would put it on with a fairly thick nap roller cover and chase with a brush.
Check for coupons and don't be afraid to ask for a discount at the paint store!