The order doesn’t really matter for the most part, but Mother 3’s callbacks to EarthBound might be more fun if you played EB first. A few parts of the game, althought small and usually inconsequential, require that you’ve familiar with EarthBound.
Besides that, each of the games feature the same RPG style combat, but have very different feels.
EarthBound Zero / EarthBound Beginnings as very much an old-school RPG. It plays like Dragon Warrior, so expect random encounters, giant maps to explore, grinding, less guidance on where to go at any time. If you get a Mother 1+2 english cart, you can get an Easy Ring which brings the encounter rate down to a reasonable level. Story is fairly light here. It’s a fun time if you like to keep things simple and wouldn’t mind having to consult a guide now and then.
EarthBound is essentially is a more modernized version of EarthBound Zero. Expect significantly more story, real cutscenes, less of a need to grind, encounters that you can see and avoid on the overworld, more mechanics, and less empty maps. There’s virtually nothing tying EarthBound to its predecessor other than the “return” of Giygas / Gyiyg. It’s extra fun if you"Get Nintendo’s official player’s guide that came with the original game":https://www.nintendo.co.jp/clvs/manuals/common/pdf/CLV-P-SAAJE.pdf and it as you play along! EarthBound’s my personal favorite, as it’s the one I keep going back to. It also has the most mods if you end up liking it enough. EarthBound is the game that “everyone and their MOTHER can love”!
Mother 3 is like EarthBound but more modern and more artsy. EarthBound has a bit more of an open-world flair, expecting you to explore the game at your own pace. Mother 3 is a series of story chapters where the game’s narrative takes center stage. You’ll also find yourself playing as many different characters in this one, not just the same party of 4. There’s also a nifty rhythm-based combat mechanic where you rack up attack combos by pressing A to the beat. This is kinda like if EarthBound was created by indie game developers with today’s technology, but on the GBA. The game’s translation feels like exactly what the game would look like if the game was released officially, probably one of the greatest fan translations of all time. Shigesato Itoi decided to get REALLY weird with this one, though, so expect maximum quirkyness and even some controversial stuff at times. I can’t agree with everything Itoi decided to put in the game, but I think there’s more than enough to make it worth putting up with. It’s an emotional rollercoster.
Overall, I’d say start with EarthBound. The original cart is crazy expensive, but you can find an EarthBound Uncut cart for about $40-$60. Personally, I find Uncut better than the original as it retains some elements from the Japanese version, but to each their own. For EarthBound Zero, I’d suggest the Mother 1+2 cart with English patch, and same for Mother 3. EarthBound might also be available on the Nintendo Switch online, but don’t quote me on that.