You might find some tips in my thread here:
https://www.canadiangunnutz.com/forum/showthread.php/1575963-Just-somethin-I-been-working-on-NEW-PROJECT
Barrel channel, set up a guide &/or jig to hog it out with a router (undersize) generally you will be able to find a round nosed bit smaller than your smallest barrel diameter. Finish it up with scrapers and chisels. You can make purpose built round nosed scrapers from a cheap $7 pack of cabinet scrapers. I usually make 2 or 3 single use scrapers per project. Will be one of your most useful tools and cheapest!!
You can do the entire barrel channel with a chisel, but that is a lot of work. you could also do it on a drill press with forstner bits... again, more work. The only way to improve on the router method would be by using a mill........ but not many of us have one of those lying around.
Action, again, router to get the gross material out of there. absolutely use a guide &/or jig. Clean/square it up with chisels, files and scrapers. Throw away the forstner bits.
Chisels, you can probably get by with Canadian Tire chisels, but it won't be fun & you will be sharpening them every other cut & they are quite a bit heavier than a carving chisel. If you decide to buy, I've tried both Two Cherries & Pfiel, I find the Two Cherries hold their edge longer and have a better starting geometry. These suckers are $40-$60 each, so choose CAREFULLY, I have about 20 of them now, but started with only what I needed & built up. Starting with about 3 worked, still there are only 6-8 I use regularly and 3 I use on every job (5mm straight chisel, 20mm straight chisel, 6/9 sweep straight gouge) Chip carving knives are also invaluable.
Sharp, sharp-sharp-sharp..... keep them sharp, save yourself grief. I strop every 5th - 10th cut and usually hone at least once a session....
Router, the more horsepower the better, you need a variable speed for hardwoods as well, also you can get by just fine with straight cutters, but upcut spiral bits will make life WAY easier (less cuts) not sure if the spiral bits are available with rounded noses. (that would be SWEET!) You don't need carbide, HSS will be just fine (and easier to maintain)
Measuring tools, precision 6" machinists ruler, one or more 3"/4" precision machinists squares, a striking knife and sharp pencils, for longer stocks a good straight edge or long steel ruler will be needed. One of my favorites & best money spent was a veritas marking guage from Lee Valley.
http://www.leevalley.com/en/wood/page.aspx?p=59455&cat=1,42936,59455 - absolutely invaluable.
Abrasives, throw out all your Canadian Tire sandpaper and never look back. Go to home depot & buy the Norton sheet abrasives .... you will be shocked at how bad CT papers are. For files, well.... CT and Princess Auto files are all single use POC - but you can struggle through with them. you can get a nice 6pc set of Norton files from Amazon for about $40 & will cover 99% of what you need. Rasps are a different story, any hardware bought rasp is gonna give you meh results and dull fast. but you may be stuck with that option as a quality rasp will start around $80 (each) I use the Auriou hand stitched Rasps ... never looked back. I use a grain 3 (special order) a grain 9 & have a grain 11 inbound. you can get by with the coarsest of grains (3) for shaping the stock and LONG you want your rasp to be at least 10" - 12" is better, as is wider. Sadly the Aurious cost starts around $150 each. (why I only have 3) There are other options though, there is a nice roundup here:
https://www.canadianwoodworking.com/half-round-cabinet-rasps
So you can easily spend more on tools than the gun is worth but in reality you can get by with a bare minimum. But crap (or dull) tools ar just going to runin your finger, work or mental stability. Like they say, "It's a poor craftsman who blames his tools" - is a flat out lie, sometimes it is the POS tool.
Plan it out on paper, lay it out on your blank and have at it... it would be best/easiest for layout/measurement if you were to both plane the blank and square up the top.. leave it for a bit to make sure it doesn't want to warp or do anything weird.
If you have spent $800 on some imported Turkish walnut blank that looks like god smiled upon, maybe consider hitting the lumber yard for some poplar and turn out a few dozen practice stocks first.
-enjoy!