Is there some video or something about what is the non-mayor player's beginning of the game like? Which are the differences to mayor's beginning of the game?
Well, you will appear at the train station without being greeted by anyone. A bit like the mayor, you still have to head to the Town Hall, talk to Shizue (or whatever English name they come up with), head to Nook Homes and finally choose exactly where you want your house to be. Again, talk to Shizue, and you're pretty much free to do whatever you want...
Wait a second... You'll have to talk to every villager in your town, compose letters (you don't have to send them), get a shovel and bury 3 fruits or sell them, catch 3 different bugs or fishes (you can buy either a net or fishing rod for 500 bells each). Otherwise, you will be stuck with the same exact amount of villagers and they won't be able to move in or out. Once you're done, you'll obtain the watering can from her.
Want to be able to use the Train Station? Well, you need to get a photo for your ID card, and it costs 500 bells. Sell things at either the shop or R.Parkers (recommended, as you gain more bells).
So, a non-mayor character won't get to plant the symbol tree, and will be never asked for becoming a mayor, since there is already one.