That would be best. The way I see it, you only have six slots in your party so you should cover as many Pokemon types & attack types as you possibly can.
Yes you can beat the game. You can expect the trademark Pokemon campaign/story here. Go town to town and challenge all the gyms, take on the League and become the Pokemon Master of that region.
The real fun is everything else.
To get Sylveon firstly you have to max out it's happiness, then teach it a Fairy type move and finally level up.
EDIT: Perhaps it will be better if you watch this:
Yup, I thought it was a nice little mode for improving your Pokemon's happiness but in truth it does more than that. I used it for about 10 minutes on Pikachu & Charmeleon and had already received XP bonuses at the end of battles.
Caught another Eevee. Deposited it on the GTS for a Charmander.
30 seconds later I got a Charmander.
Now I'm going to deposit the Charmander for a Bulbasaur.
Eevee is the golden ticket.
EDIT: Got a Bulbasaur.
Added @Ramar & @The Peeps
I did the exact same thing, so far I've managed to paralyse everyone except for Emolga so I'm thinking that Electric Pokemon can't be paralysed anymore, if that's the case then can fire types get burnt/Ice frozen/Poison poisoned etc?
A little high I'll admit
Part of the reason why my party are at 22-25 (same stage as you right now). but you can fix that by just using a Pokemon that can paralyze, coupled with capture o-power = good times.