yeah budget here is going to be your factor
I started shooting 1,000 yard matches back in the late 80's
how serious do you PLAN to get, as its sometimes cheaper to cry once and buy once, than buy something now and then want else later on
about any basic center fire rifle caliber can get there, some just do so better
a 300 win mag is a good basic get you there with LOTS of proven info out there for it to help you learn
being a lefty isn't a big deal in LONG range work as most things shots are NOT in a rush, so loading one at a time and even if you have to use a right hand bolt, isn't the end of the world,
which brings you back to $$/budget
have the funds a rifle can be built for you any way you want it
\ if you go to any 1,000 yard match, you will hear a LOT of guys say this caliber is BEST or THIS one is better and blah blah blah LOL
people, will always like what they like and push it
YOUR the one that is going to be buying and paying for things, BUY what you want, want suggestions on calibers, read some match results and calibers used to get there scores
you will, quickly come up with a short list of the most scoring calibers
and then decide on what fits you better
recoil, HOW easy or not getting ammo will be(brass and such) you can then look at costs differences in calibers too(heads brass) and go from there