Having owned or currently own four of the five Cals that have been mentioned with factory loads avail, my call is for the .223 based on you don't reload as of yet.
Of the three the OP is considering i can comment on three from experience and just so they have for future reference (I added in the 204R)
.204Ruger
Don't let the .20Cal designation fool you. This is a capable cartridge up to Coyote. It really shines if you reload in the 35gr-39gr window however. Has a long case & barrel life even though it can push the 38-4100fps realms with reloads (longer than 22-250 would last). From Crows, Groundhogs to Coyote this lil guy can do it with minimal recoil & noise. Something to consider. One of the bonuses withe .204 is it's low recoil & noise. Once you touch off a round there's barely any recoil & no loss of sight target wise. Get ready to watch pink mists through the scope if your hunting with this Cal
.223
Largest selection of factory & after-market bullets available. Depending on the twist rate, you'll be able to run from 50gr-80gr projectiles. No worries about barrel life here. Most manufacturers have a wide selection of .223 Varmint & regular configurations avail, your sure to find something that you like. A great Cal to start with for the Precision/LR hunting arena.
22-250
Once the most popular varmint cal, now it's taking a somewhat backseat to the .204 & .223 combined. Flat & fast, it gets beat out by the .204R beyond 3-350yrds trajectory wise. Barrel life could be a problem if you intend on high volume shooting however. I traded my 22-250 based on this.
.243
6mm/6.5mm have some of the best BCs in projectiles bar none. Accurate and has punch well beyond 400yrds. Don't believe it? Search youtube for "dartman76" user and watch his neighbour shoot out from 5-1000yrds on Coyotes with his .243 Overkill for the smaller stuff, maybe not enough juice for exceptionally large deer as some would say, it still should consider a serious look.
6.5 Creedmore
I looked long and hard at this Cal 2-3yrs ago when it came out. Brass and factory ammo was only avail by Hornady then & everyone was still just starting to mess with it on the loading bench, so there wasn't much info available just yet. If i was out west coyote hunting I might consider it as an ultra long range cal based on the above noted BCs. Then again, it's nothing a 25-06, 243, 6.5Swe or 270 couldn't handle with more readily avail factory ammunition. One might eventually find it's way into the cabinet someday as it intrigues me
As for Savage, Remington or Tikka debate? Pick one, they'll all do what the OP is looking for.