unless you want to challenge yourself on this playthrough, i would not suggest keeping only one pokemon. i don't know if you've played any other rpg games, but you typically have a team and you want to keep all of them in good shape and level. each pokemon is going to have its strengths and weaknesses. for example, if you started with charmander, you're going to have a difficult time taking down the first gym leader with only that. i would recommend catching a caterpie and training it until it evolves at level 10. it's well worth the effort, though it will seem fruitless at first. i recommend catching lots of different pokemon and trying them out to see how you like them and to find a balance in the types of pokemon on your team. for example, you don't want to go around with a team of only water types. it's doable, but if this if your first time playing, i would suggest more variety. it will leave you better equipped for the challenges ahead.
as far as leveling goes, my personal preference is to be at the gym leader's level or possibly higher before taking them on. in the case of the first gym leader, that would be around level 14. again, not necessary, just my usual protocol.
you may only carry six pokemon on you at one time. additional pokemon will go to the pc storage box (someone's pc) and you can withdraw them from there, as well as deposit any pokemon you have that you want to stow away for a while. boxes can hold up to 30 pokemon. when your box is full, you will need to go to the computer and hit 'change box' to select an empty box where pokemon will be stored then. you cannot catch wild pokemon if your pc storage box is set to a full box. for this reason, i tend to switch my box to the next empty one when my box hits around 25/30, just so i don't accidentally go hog wild catching pokemon and then run out of room quickly.