Most of your beneficial bacteria should be in the filter media. This is what you want to transfer. If you have a filter that you can take the media out of (sponges, bioballs, ceramics etc) then it should be a simple case of transferring the media to the new filter. If you have to cut the sponges so they fit, it won't matter.
As time goes on and your tank becomes established (where some of the bacteria grows in a biofilm on the entire surface of the tank, substrate, ornaments etc), you can look at slowly changing the filter sponges to the ones that are designed for the filter if it means enough to you. After a few months the tank should be mature enough that replacing one sponge every couple of months shouldn't affect the bacteria colonies too badly.
Could you let us know what filter you have now, and what you might be looking at as a new filter? It would be good to know the type (hang on back, canister etc), the make and model (e.g. Fluval FX5 or whatever) and what's in your current filter (Sponges, bioballs etc).
