Veneering isn't too tough, although finishing speakers with a near-flawless finish is easier said than done. The problem you're going to face is that your speakers are already has a finish on it. Meaning, gluing another veneer on top of it will be tougher and potentially more troublesome (air pockets) than if you were working with unfinished wood.
You can get veneer, either standard or paper backed. The paper backed stuff is thicker and somewhat easier to work with if you haven't veneered before. The paper backed stuff is also more expensive, but I would recommend the paper backed stuff in your case.
You'll have to try and remove as much of the original finish as possible by sanding. Then, you can use contact cement on both the cabinet and the veneer. THen apply the veneer and trim the excess with a veneer saw, sharp razor, or with flush trimming bit and a router. Since veneer is VERY thin, you won't need to do much final sanding. You'll also have to trim out the speaker cutouts as well. then comes the finish, you're choice here of what you want, gloss, semi-gloss, film vs. oil, etc. This will be mostly personal preference and whether or not you want to spray vs. brush vs. wipe on. There's oil/varnish, lacquer, shellac, polyurethane, or some other mixture.
If you're uncomfortable doing any of this, take it to a professional cabinet shop where they can veneer it for you OR you can make another set of cabinets with MDF and put your own veneer on. This way should something go wrong, you're not left with irrepairable damage on your original speakers.
