I just finished building a .300 Ultra Mag on a Stevens 200. Still doing load development and have yet to drive it at the ranges that it's meant for but so far I've got a load that puts 3 bullets in 1 hole at 100 yards!
First thing I'd say is look at screwing on a new barrel. I've got a Savage .270 that's complete factory that will shoot clover leafs all day but if you're looking at competing, you probably want a thicker and longer tube. Also you may find you want a chambering with less recoil for competiotion. If you're getting a new barrel anyway, a $25 bolt head would be all it would take to go to a different chambering like a 6x47 or 6.5/284.
Second thing would be the stock. I haven't shot one of the new accustocks yet so I can't comment on those but I know the older ones have kind of a tupperware feel. Again this isn't something that would bother me in a beater hunting stick but a comp gun is a whole different ball game. Check out Stockade, Mcmillan or MPI
Third would be the trigger. The accutrigger is great in a hunting gun but if you're looking at competition shooting I'd go for something like a Rifle Basix.
I picked up a nikon buckmaster 6-18 with a mil dot at midway USA for $300. I think they still have them on sale. They also have the barrel wrench and barrel vise that you will need to switch barrels yourself.
Good luck and keep the pics coming!