Usually I stick with my starter untill after I beat the elite four. Some games I stick with my starter after the Elite four and some games I stop using it Because there are other Pokémon I want to train and use.
I always stick with my starter until I get to the point when I either A) leveled them up to level 50 (pre-Gen 4) or decide to train Pokemon for the Battle Tower/Frontier. Even when I have a weak starter, I picked them for a reason, so I keep them through thick and thin.
Well, for the most part yes, I keep my starter on my team for like ever... However there is one exception to that ... in Soul Silver... I got to one point in the game and when like... you know what? I am going to have a whole freaking team of eevees and eevee evolutions .... yeah...
Now its a team of level 80 plus of just eevees... hehehehe
I always stick with my starter, I mean it is the very first Pokemon you receive in any game; it is the only one that is with you the entire game (Unless you place it in the PC, trade it, or release it).