Should I filter my beer before bottling?

I have a fine mesh stainless steel sieve that I've used to filter out the trub from the boiling pot to the fermenter. My question is, can I run the siphoned beer from my fermenter through the sieve while I fill the bottling bucket? I know there is talk about not wanting to oxygenate the beer before bottling. I suppose it's possible the beer will pick oxygen while going through the sieve. Is this a good idea or bad?

I doubt its a bad idea.

However, since you've already run it through the mesh once, unless you used a flocculating agent, almost everything in your fermentor is capable of passing back through it into the bottling bucket.

You're right about oxygenating through the sieve. Bottle it, condition it, and let it sit in the fridge for 2 weeks and it will be very clear. You'll be even better off if you used some kind of fining agent along the way (Irish moss, whirlfloc, gelatin, etc.).

If you want to filter your beer, you need a a couple of kegs, a plate or cartridge filter, and (of course), CO2 to push the beer through the filter between the kegs. A strainer, nylon bag, etc are way too porous to filter out most hazes, and you'll introduce oxygen into the beer, potentially reducing its shelf life.

If you use Whirlfloc in the boil, most beers will clear just fine by itself within a month or so. If not, you can use various finings to improve the clarity of the beer. When siphoning, be sure to minimize trub and yeast pickup - that tends to help as well.
Colder temperatures tend to accelerate the flocculation of yeast and help settle various particulates - so if you have the ability to cold crash or lager your beer, that's another option you can try.