Oil based poly, especially the typical brands at the big box stores, is way too thick to be applied by brush by most of us mere mortals. And stretching the finish over a larger area in order to obtain a thin coat exacerbates the problem even more. It is just too thick to level on its own if you over work it even slightly.
Even though the can will likely tell you not to thin it, thinning with mineral spirits or naphtha even 10% by volume will vastly improve its ability to level on its own. Naphtha flashes off faster giving the film less time to collect dust nibs, one cause of the bumps you talk about.
While you can buy all sorts of products out there like stain preconditioner and sanding sealer, a thinned coat of poly (or what ever final finish you will be using) will serve basically the same purpose. Sanding sealer is basically polyurethane with some stearates added that help prevent clogging of sandpaper in subsequent sandings.
Grain filler mentioned previously is not that putty stuff at the big box but is generally a thick liquid available in different shades which is applied to the wood and worked into the grain with the excess being squeegied (is that a word?) off. Once it has dried, light sanding removes the remainder of the excess from the surface. A couple applications may be required if you have really deep grained wood. You can also fill the grain by using some poly as a lubricant to sand the surface creating a slurry which can then be worked into the grain and allowed to dry in the same manner as the grain filler. You can also fill the grain by applying numerous coats of your film finish, then sand back to the wood between coats.
Another option for a sealer for the wood, particularly useful if you are staining, is a spit coat of dewaxed shellac. I like using shellac on oak and other woods susceptible to grain raising, to intentionally raise the grain, then sand using a sanding block and 320 or 400 grit paper. This is particularly good with hardwood plywood where you have concern over sanding through the veneer. That is the only sanding I generally do on hardwood ply.
Concerning the use of grain filler and/or stain you can obtain different looks quite simply by the color of grain filler used and the order of application. For example you can stain the wood, then apply a grain filler similar to the stain color for one affect, of contrasting color for another. You can use just grain filler of a contrasting or similar color to achieve other affects. Or you can apply the grain filler then stain, again obtaining different affects by the color of the filler in comparison to the stain. You can also condition the wood by applying a spit coat of shellac at some point in there to obtain even other affects. The dewaxed shellac also makes a great barrier between your stain/grain filler and the poly or other filming finish.
Get you some scraps, and some of the stuff mentioned and play around with it. And thin your poly so it flows better and self levels and have fun.