I've been reading about paleo diet and it seems that I would get the nutrients I need, but it has a basis of no dairy and no grains. Is this ok?

Good Analysis. Reverting to a 'caveman' diet makes you prone to caveman diseases. There are definite advantages to a low-carb, low fat diet, but grains and dietary sourced of calcium are needed. I suggest you look at the mediterranean diet.
Paleo diet. Yes, it is fine to eliminate those food groups from your diet and many people feel better without grains and dairy. Although many do well on them as well. Diet is unique for everyone and any change should be monitored for is benefit and any potential harm.
Yes. Grains , legumes and dairy are not part of a natural human diet. Our ancestors survived and reproduced just fine for 200, 000 years without these neolithic foods. Grains legumes and dairy have been introduced to the human diet over that past 10, 000-500 years (depending where you are form). What followed was the decline of our health. Jus tlook around!