Absolute best way to earn exp as of 5/1/2019 is to play the team modes/make it count.
FFA and most wanted are decent but i always quit out and search for a new game because i find most wanted really annoying and FFA is highly skill dependant so if youre good, it can yield decent exp.
As for team modes, you can earn 250-350 exp if youre on the winning team.
Make it count is what you want to focus on. Its basically a dumbed down battle royale mode. Ive won 95% of all the make it count games ive played by simply going clockwise or anti-clockwise close to the perimeter of where youre suppose to be and where you arent allowed to be. In other words, stay near the outer ring where its greyed out but dont step outside as youll automatically die if 10 seconds pass.
By doing this, youll buy yourself time, letting the other players kill each other, leaving maybe only one or two players allowing you an easy win. *make* sure you have vibration on as any player nearing your proximity will make your dualshock 4 vibrate. If youre not confident in killing off the few remaining survivors, simply stay alive (you dont have to win) to gain a ton of exp. You can easily earn between 300-500+ exp by either winning or staying alive. If you stayed alive long enough and killed a few but died, dont quit! Youll still be rewarded but only between 200-300 which is still decent. Also dont stand still at any point as you will show up on the other players minimap and always crouch and move slow.
Be prepared for a grind as going up just one level can easily take a few hours depending whether or not youre on the winning teams or not or how much you manage to win in make it count.
Side note: how much exp you actually get is based on how much time has passed in a match of make it count or any team mode. If you finish make it count in 2-3 minutes the reward is greatly reduced. However, if 8 minutes has passed for example, the reward is much greater.
The exp range i provided above still holds true provided enough time have passed in any given match.
A good example of this is the team mode where you need to capture zones. If you manage to capture all zones on your team relatively quick the exp reward can be as low as 25-50 exp. So there is absolutely no incentive to quickly winning a match in regards to rewards.
Edit: i had to change the amount of exp you earn as rockstar has apparently made a hotfix that rewards you even less now than
before. It used to be between 600-800+ for make it count for winning/letting enough time pass and 300-500 by simply staying alive long enough but not doing so well (dying/not killing anyone) etc.