I've done several guns over the years. I used wet/dry sandpaper on the metal parts. I recommend starting with 220 grit and working up to at least 800 or 1000. That should give your bluing a nice lustre when finished.
As for brand of bluing to use, I'll leave that suggestion to some of the other guys on here as I wasn't really pleased with the last stuff I got at Bass Pro. Can't remember the brand but its in a blue plastic bottle. One thing I will mention though is that degreasing/cleaning the metal parts very thoroughly before bluing is critical to getting a good result.
As to the wood, I've always carefully scraped off the old finish and then sanded with progressively finer sandpaper going up to at least 180. Here's a tip, once you feel the stock is nice and smooth after sanding with say 120 grit, wipe off the dust and then dampen the entire surface with a clean wet rag. This makes the tiny wood fibers stand up once the wood is dry. Then sand and repeat a couple times with finer grit paper. This will help any grain stand out a bit better and give you a smoother finish in the end. It doesn't take very long to dry each time, a day maybe if kept in a reasonably warm dry spot. Just don't soak the stock and then put it next to the wood stove as you might cause cracking. Unlikely but no sense taking the risk. I've used urethane on some stocks and linseed oil on others. Urethane will shine more than an oiled finish usually but either works well.
Having said that, the stock doesn't look to bad in the picture. If it isn't too bad you might want to just keep the original finish. I most cases you will find that the wood is light coloured, maple or birch or birch maybe, under the existing finish. I've found that its hard to get back to the original finish colour using stain but some of the guys on here may know of specific products that work well.