Sand the edges with 150 grit but do not go anywhere near the veneer with 150. It is very thin and you will go right through it. Then sand the whole thing (including veneer) with 220. Do not go finer than that as it will close up the pours in the wood.
If you insist on staining it, you will need to put a coat of sanding sealer on it first. Birch gets really blotchy if you try to stain it without a sanding sealer. Personally, I would either just go with straight varnish or if you want to bring out the grain a bit, you could do a coat of boiled linseed oil first.
Then, use a wipe on varnish. Trust me on this. The old school paint-on varnish sucks to work with and trying to do a flat surface without leaving brush marks is tough. This is my secret weapon for finishing:
https://www.wwhardware.com/general-f...fault_products
You put it on with just a clean rag - no brush at all. An old tee-shirt works best. Wipe on two heavy coats letting them get completely dry. After the second coat dries completely, sand the whole thing with 320 grit, then do two more lighter coats. Don't let it scare you. It will look like absolute shit until you do the 4th coat; then it will look amazing.
You can be done after the 4th coat, but if you really want it to look like you can put your hand right through it, sand it again with 400 and do one or two more light coats.
This stuff works great because putting it on with a rag instead of a brush, you do not get drips or run marks. The downside is you do have to do at least 4 coats. With traditional brush-on varnish, you can usually get by with 2 coats (plus fixing all the damn mistakes and touching them up).