That's a very nice bowl for your second attempt Tony! For finish, it would depend on what you wanted it to look like, I find finishing oil looks very nice on bowls and has the added benefit that you can apply three or four coats and buff it to a gloss, or leave it at a couple of coats and cut it with a nyweb pad to a nice satin finish. You need to leave it a day or so to fully dry. If you wanted a very glossy finish you could try acrylic sanding sealer cut back and then two or three coats of acrylic gloss (chestnut do these in rattle cans).
Not sure what chuck you have, but it looks like an expanding collet type, bowls can be made finer with a less bottom heavy profile if you have an external dovetail that the chuck clamps onto, rather than an internal such as here. Nothing wrong with this bowl mind, just pointing that out.
Hah, Ian beat me to it! Ian is right about the wood too, it is Beech.