Lots of good guidance here...

.338 for less than 1000 is a waste. It you only want to shoot to 1000, get a .300WM and call it a day.

As a starter, the Savage 110FCP is about the cheapest you are going to get into one. They are good shooters, but they can have a chamber issue if you get an early one. I wouldn't bother with the more expensive Remington MLR, it won't be any better of a shooter than the Savage, and is 40-50% more expensive.

The Sako TRG 42 is a great rifle, that is very accurate. There has been reports of weak stocks on them, and even a few people that have broken their stocks on them. If you intend to use the rifle in the field, matches etc, I would suggest upgrading to the KRG offerings.

Customs, off of the Badger action or Surgeon actions, would be a good option, but I would be willing to bet your going to see a 9-12 month wait for a custom. They will be hella good shooters.

A DTA would be a good option, and would allow for multiple calibers to play with for less than multiple guns. They have proven to be very accurate, and the couple of times I have shot them, they are a pleasure to shoot. One of the guys at Trigger Time has worked a few triggers on these and they get even better than they started out. Still not a perfectly tuned Jewell or otherwise, but damn good for a bullpup.

And of course, there is the AI. Hoser would be a much better resource about these, but the one that I have gotten to shoot was great. There is a reason they are considered the top tier. They are exceptionally accurate, and they just run. No matter what.

Bottom line, it's not a cheap round to get into. The rifles can be reasonable to extremely expensive, the glass necessary to keep up with the rifles is expensive, and the ammo is expensive (less so if you reload).

All that being said, it is a fun round. If your budget allows for it, and you want to start shooting distance, then by all means, go for it. You'll enjoy it.