So when I do buy a rifle I should try different rounds until I find one that shoots well in it? Or are there specific brands that are good?
IMHO the Lee Enfield is a great rifle. Many of them were purpose built as sporting rifles, on new receivers, both by the manufacturers of those receivers and by many cottage gunsmiths of varying ability.
Later, after many wars and hasty manufacture, many militaries dumped their Lee Enfields onto the surplus markets, where they were either picked up by other nations for their military or sold off to civilians.
Most of those sold to civilians were "customized" in various and imaginative ways, which mostly buggered the ways (bedding) if it wasn't already bad.
One thing that many nimrods did back in the day was to toss the King Screw pillar, or just lost it. When the stocks weren't properly coated with Linseed Oil, they shrunk and hard use often made them sloppy.
So the first thing a novice would do is get out his trusty/rusty "screwdriver" and crank as hard as he could on the King Screw, crush the bedding, sometimes to the point the magazine couldn't be inserted to lock in place or couldn't be pulled out. The accuracy was usually buggered.
Before you purchase a rifle, get someone with some "real knowledge" on what to look for to come with you, and look the rifle over.
When they're set up properly, either sporterised or full wood, they will shoot most commercial offerings acceptably, but don't expect miracles.
If you aren't going to hand load, then you're going to have to purchase different makes of ammunition, as well as bullet weights, to find which shoots best in "your rifle"
Now, here's where it gets even trickier, the brand of ammo you find that shoots best will likely not shoot best or even well the next box you purchase.
Why? Because it's from a "different manufacture lot" and the case, primers, powder, bullet manufacturer may all be different, and there will be nothing on the box to indicate this.
Remember, these rifles are most well used and the newest models were made 50+ years ago in Pakistan, on used Maltby tooling.
All commercial ammunition, that I've seen over the last 20 years, for the 303Brit is loaded with .310-311 diameter bullets.
If you have a rifle with a larger nominal bore diameter, you will be lucky if anything shoots better than a 4 inch (10cm) group at 100 yds/meters.
Think long and hard before purchasing a Lee Enfield as your first rifle.
Personally, I've had a love affair with Lee Enfields and the 303 British cartridge for over 60 years. It's the reason I started hand loading, back in the day when hand loading was still considered by many to be alchemy or sorcery, and only for those who wanted to blow up a rifle in their face.
For the most part, these rifles were made for military purposes, accuracy needs were for a different purpose than those you require to hunt.
The militaries of the world are much happier wounding the combatants of their enemies of the day, than they are in killing them quickly.
A dead combatant can be left for later retrieval, or as a field flower. A wounded combatant needs to be taken care of, and depending on the circumstances, can play hell with the psyches of the other combatants at the most critical moment.
A hunter's ultimate goal is a "one shot, drop in their tracks kill"
Chasing down a wounded animal is not "fun" and often puts many hunters completely off hunting forever.
Answering your question with a quick yes or no, just isn't possible, because of how many variables there are with this particular platform, we call the Lee Enfield.
There's a very good reason people LOVE TO HATE LEE ENFIELDS. Some of it is the fault of the platform, and some of it is the fault of the shooter.
A Lee Enfield is the least desirable platform to start a newbie on IMHO.