Yes you can but you miss out on the good online parts and DLC.
Mario Galaxy, Brawl and Mario Kart are all must-haves. Mario & Sonic at the Olympics I think is one of the better party games out there (and knackering on the arms).
If you want some on rails shooting, you could do worse than the recently released House of the Dead trilogy or Resident Evil Umbrella Chronicles. I understand there's another House of the Dead about to come out too which might be better.
I understand that No More Heroes is good too, and those of us over 18 are also patiently waiting for Mad World. I can also recommend My Life as a King on the Virtual Console for something a little different.
Incidentally, I hate Wii Sports and it hasn't been in my console since the day I got it. It's just too easy to be fun in my opinion.