Welcome to LRH! A lot of good guys here with good info. In my opinion, if you want to start out on a budget Savage has your back. They just came out with two different models in 338 Lapua that will get you started and a base to build off of as you gain experience.
Here they are:
Savage Arms Firearms > 111 Long Range Hunter (338 LM)
Savage Arms Firearms > 110 FCP HS Precision
Of the two of them, I would get the FCPHS Precision. It gives you an HS stock which is already after market and a great foundation.
As far as scopes are concerned, definitely spend the coin on a the best possible glass you can afford. I am an avid Vortex fan and like the PST. Though I admit I don't own one, its next on the list to buy. I do however own a Vortex Viper. Customer service is great if you have any questions.
The rifle already comes with a base, so rings are the next to buy. If you must have top of the line, then IOR or Night Force would be the way to go. Leupold would be my second option. I currently own Weaver Tactical but wouldn't suggest them for the 338LM. I will suggest the Burris XTR Tactical rings. I have owned them and I am still kicking my self for getting rid of them. Warne makes some really sturdy rings also.
Savage is a good base platform for the do-it-yourselfer. I built my own rifle off of a Savage action and its nice to see they make one in a long action suited for the Lapua Mag. You could only do one better and that is the BA 110:
Savage Arms Firearms > 110 BA From my understanding this is essentially a BAT action, with a PDC stock. The nice this is that you can change to the grip to any AR style grip that you want. It does cost a lot, but about the same as a custom and you already get a BAT action and a PDC stock.
These are my suggestions. I know other guys will give some great advice on other builds.
Tank