Most speakers have built-in crossovers, which means you feed a full-range signal to each speaker - i.e., your run a pair of wires from the L channel on the amp to the left speaker, and a pair of wires from the R channel on the amp to the right speaker - and the crossover in each speaker takes care of sending the higher frequencies to the tweeter and the lower frequencies to the woofer.
So, in short, no, you won't have the same issue as you do with your current speakers.
Your budget will determine what speakers you buy, but:
- if you buy cheap, you will likely want to upgrade before too long, so don't hesitate to invest in better main (L+R) speakers;
- if you buy bookshelf speakers, you will almost certainly require a subwoofer to handle the lower frequencies.