The 25-06 is a fine caliber for deer sized animals and even elk with the right bullets.  The .270 is a round that will never die.  You can load it up or load for large game or down for the smaller animals and the 7mm-08 is right in there also.  One that you didn't mention would be the .243, it has killed a lot of deer.  Lot of help arn't I 

If it was me I would go with the .25-06.  For some reason I just like it. 

My favorite gun caliber is the .270.  I've used mine for pronghorn, whitetail and mountain goat.  Powerful, but pleasant to shoot.

Probably overkill for coyote so if you're looking for a deer/coyote gun it might not be the one to choose.

I like my .243 too.  Taken coyote, whitetail and pronghorn with it too.  No recoil so it is a easy gun to shoot and practice with too.

I don't know much about the .25-06 so cant help you there.  I'd probably look at the ballistics in terms of drop and power.

What game are you going to hunt the most and at what distances?  Will the caliber have sufficient power to reach and touch them at the distance you'll likely shoot?

Truth is while this is your first rifle, it likely won't be your only one so you might factor that into the equation.

deer would be under 200yd shots, coyotes maybe that 300 yd range...

.25-06 is fantastic for deer and coyotes.  However, the Marlin X7 is a better value.  Costs about the same but higher quality rifle than the Axis.

All are great for coyotes and deer.  For your purposes, I like the 25-06 the best. It is such a flat shooting round.  That being said, they all would work great.

You should consider availability of ammo.  Depending on where you live, 25-06 might be harder to find than the others. I can walk into any Walmart here and find 270, but usually not 25-06.

Also do you plan on only hunting deer and coyotes?  If you have plans for larger game (ELK) in your future, you may want to consider the 270.

My .270 Win is my primary go-to biggame rifle only because I hunt elk in addition to deer and pronghorn.  If I only hunted deer and antelope then my go-to rifle would likley be one chambered in .25-06.  I think if you hunt deer and shoot coyotes out west then the .25-06 is a great cartridge for that combo of hunting/shooting.

I know you can load light bullets in a .270 Win for varmit, but why bother?  There are just too many lighter cartridges out there that are better to shoot varmit with......such as the .223 Rem, .22-250, 220 Swift, .250-3000 Savage, 6mm Rem, even the .25-06.  But of those I categorize both the .25-06  and the 6mm Rem as going into medium biggame cartridge terrory.

I shoot a .243 Sako, a 25-06 Ruger, and several different 30-06s.  If you stay strictly with the animals you mentioned in the OP then I would go with the 25-06 only because it will reach out further than the 243.  It will take either of those animals at 400 yards and beyond with no problem with the correct bullet.

If your not going to reload, look at ammo availability and cost in your area.  Chances are good that the 270 win  will be more available and cost less and should be a consideration in your choice.  If you reload go with the 25-06 which will have a better range of bullets for your application (hunting coyotes and deer).