Definitely look at the various Remington 700s. Their SPS entry level rifles are fairly inexpensive and available in a wide variety of calibers. They are a great platform to start with as everyone makes aftermarket parts for Rem700s and all gunsmiths will work on them.
If you want to go a little higher end to start, the earlier suggestion of the Sendero is a good one.
As an example, here's my nephew's Rem700 SPS....
Nephew's Transformed Rifle!
If you like a more dedicated longrange rifle, here's an example of an SPS Varmint....
My 308 1k Rifle - The Evolution Continues!
Read all the way to the end of the thread and you will see this 308 turn into a full-blown custom 260.