Hopefully this doesn't become too long winded....
Firstly, define what you want to be as "long range", of course, also base that off of what sort of distances you have access to on a regular or semi-regular basis. For some shooters, 300m is "long range". That being said, I suspect you want to go a lot further than that.
From a cartridge perspective, pick something that you can get factory loaded. If you aren't set up yet for reloading, it's a great way to accumulate brass, and an opportunity to get to the range without having to invest all the money, necessary tools and components to re-load ammo. Reloading truly is another ball-game, and might not be something you want to mess with immediately. You're idea of 6.5 is excellent - while ammo availability is a bit spotty across the board right now, there is generally plenty of 6.5 Creedmoor ammo, in varying weights, quality etc. The same goes for .308. Being that this is your first foray into long range, having a good selection of factory available ammunition will help immensely. Even a .223 can be an excellent choice.
Plenty of people will give plenty of suggestions for actions - from factory rifles with varying degrees of quality (like a rem 783, made by a company who barely exists) to the custom actions. Personally, I have a bunch of experience on Bergara rifles, both the B14 and a Premier rifle, and I am extremely happy with them. They offer excellent factory options, and given that they accept a large majority Remington 700 parts (triggers, bases, chassis etc.) you really can't go wrong with them. I equally enjoy custom actions, but as your first foray into long range, starting with a known quality factory rifle will give you tons of room to grow, and figure out what you do and don't like in a precision long range rifle. It can absolutely be overwhelming when you start to see all the accessories, products, etc. out there for long range shooting.
Take a real look at how capable you are of shooting accurately and and consistently on a regular basis - most people getting into long range for the first time wont shoot consistently enough to necessarily warrant swapping the barrel out for a "match grade" barrel off the get go. There is merit to a match grade barrel no doubt, but to think you should need to drop the factory barrel off a rifle right away is likely not required.
I'll also throw this in. Shooting a .22 out to 200 and 300 yards is insane amounts of fun, and will teach you a ton about the fundamentals of long range shooting, especially wind reading. You may want to consider the option (at some point in time) of having a similarly set up .22 to your centre fire long range gun. Bipods, tripods, bags etc. will all be shared between them as well. If you don't regularly have access to the distances you want to shoot with your future centre fire rifle, the .22 will get you out to the range to continue building your fundamentals.
Obviously this goes without saying, but figure out your budget for an optic as well. There's plenty of great choices out there depending on what sort of long range shooting you want, and even more reticles to choose from.
