All the trim too? Dark or light stain? What style is your home?

Can you post any photos? You can upload to a free site like photobucket, resize, and paste the HTML code here. When you preview the photo should should up if you did it correctly.

You could paint them, but you'll see the grain if you don't fill it. Seeing the grain isn't necessarily bad, but not everyone likes it.

Proper prep and good quality paint and brushes will make all the difference in the paint job. A few suggestions: I like Zinsser 123 primer. BM Satin Impervo (it's oil based) paint. Just keep in mind if you choose a white it will amber slightly over time because it's oil based. Cabinet Coat is also a brand that has high ratings. It's self leveling and you can have it tinted. There are others, but I wouldn't use something cheap.

Sure you can paint but before you do just know there's really no going back for you or a subsequent owner. Stripping paint is always possible but it's a big job. That said.....
....I painted the oak vanity in one of our bathrooms. I chose not to fill the grain as Sheesharee mentioned because I don't mind seeing it. I lightly sanded, did not prime, painted with Benjamin Moore Regal Latex Pearl Finish, no finish coat. It has held up perfectly and this is a bathroom that was shared by my sons.

I painted my oak kitchen cabinets and am very happy with the results. I filled the grain with ready patch, and that is time consuming but gave the smooth look I wanted. I used 123 primer and C2 cabinet paint. Oak is one of those woods that used to be everywhere in the 80s and 90s but has really fallen out of favor in home design...which probably happens with everything, let's face it. Maybe it's be back "in" one day but I believe in decorating your home to please yourself in the here and now.

If you do paint, please post pictures! I have been wanting to do this to my oak kitchen cabinets for at least 5 or 6 years now and keep putting it off. I'm kind of dreading the work but looking forward to the end result.

We were planning on replacing them but have recently made the decision to sell the house in a few years, so a full kitchen remodel won't be happening. Just replacing the appliances, painting the cabinets, and installing hardware.

Seeing your cabinets might inspire me to finally paint mine!

We did this in our home.

Before (with new counters, appliances, and backsplash)

After

We had pro painters do the rest of the house and I used the same primer and paint they had used. SW's best cabinet primer, and their new alkyd cabinet trim paint.

Process was to lightly sand, then wash with TSP and rinse. The using brush and small foam rollers for cabinet and trim, paint on the paint with the brush, and lay off with the roller so no brush marks. Did 1 coat primer and 2 coats paint. Has been very easy to keep clean. I will do minor touch ups about twice a year, which is just dabbing any dings with a foam brush and paint.