You can buy either online or cartridge. If you are looking to save money, used cartridges will likely be the best bet as Pokemon games are rarely on sale. I tend to buy one version physical and the other digital, but I often pick up a few physical versions later. I own 3 switches (an OLED, normal Switch, and a lite. My nieces and nephew comes over and we will often play together).
You can catch pokemon without battling them by simply hitting them with a ball. Pokemon have different "moods" where they will either be chill around you, run away, or attack. You are able to craft pokeballs, potions, and other items. It is semi-open world. There are five major areas that are open where various pokemon to catch. There are very few battles in the game as most of it is about quests and catching pokemon. There aren't gyms, instead your story pace is tracked by quelled Noble pokemon which you fight mainly instead of your pokemon as well as your star rank in the Survey Corps. You need to dodge their attacks, hit them with balms, and battle them occassionally.Also how different is Legends Arceus from the rest of the mainline games? What is so different?
Let's Go feels like a dumbed down Pokemon Yellow. You get either Pikachu or Eevee (Version dependent) as a starter that has about 100 BST increase over their normal. There are pretty much no wild pokemon battles (you do fight the Legendaries and Snorlax but catching is still the same). Catching uses Pokemon Go mechanics with one joycon/pokeball controller if you have it.Because pokemon games entering that 3D era made them very much different from what i remember when playing them on GBA.
Say, Lets Go has the same feel as old (gen 1, 2, 3) games, then theres something like Sun and Moon and also Sword Shield (correct me if im wrong) that have fixed camera but the area isnt that 'boxxy' like Lets Go, and then theres Scarlet Violet which has the camera in third person, behind the players back.
I have a feeling the last two types have less of that open world puzzle element than Lets Go and first few gen games, because that was a big part of the gameplay, at least for the first time of playthrough, beside just pokemon battles.
BDSP is basically just Diamond and Pearl with a chibi overlay. The only differences are the underground where pokemon can spawn and the fact the Elite 4 is harder than based BDSP as they have max IVs and some EV even on first round.
Legends and ScVi are the only ones with a free cam in all cases (save cut scenes). All others have fixed cameras (save SwSh in DLC areas and Wild Area in base game.)