buy the cheapest Rem 700 you can find, then drop it into an H&S stock, all together you are still less than $1,000. throw some rounds down range and note your progress. Once you can see some steady improvement you have 2 options.
Option 1: send the rifle to either a local gunsmith or one of the sponsors on here for truing services. They will true the action, true the bolt, true the barrel, bed the stock and boom for about $700 you are now a 1/2moa rifle.
Option 2: Send the rifle to a gunsmith for a new barrel. New barrel will be about $350 and you will still have to pay for action truing, your new barrel will need to have tendon threads, chamber, true to 12oclock, cut to length, possibly flute and thread and all that costs money. for all of this this you are looking at close to or over $1,000.
If you choose to change out the stock consider your true purpose before purchase. Some are great but heavy as hell, some may have options you do not need, and some may be out of your pay range. If you want a drop in aluminum you have Rock Solid at $750 or AI at $1,300. If you want something lighter you have Manners and McMillan. If it is just for the short range you mentioned a properly bedded H&S is just fine.
BEING YOUR FIRST RIFLE UPGRADE ONE STEP AT A TIME, DONT WASTE YOUR MONEY ON THE "GOOD" SHIT TILL YOU GET PLENTY OF TIME BEHIND THE TRIGGER AND GET A BETTER FEEL FOR WHAT YOUR LONG TERM GOALS ARE WITH SAID RIFLE.
P.S. My light weight deer rifle is a Rem 700 AAC-SD that was trued and bedded in an H&S, it holds 1/2moa and in November it took a buck at 200yds with a heart shot.