There are a good number of non-combat activities in EVE of which the main options are exploration, salvage, hauling, mining, industry and market trading.
For industry and market trading you need no ships at all (although a quick frigate to move from station to station or structure will be helpful).
For exploration of relic and data sites your best choice is a frigate, although there are good cruiser options for this as well.
For salvage the ‘best’ ship for this purpose is the Noctis, which is a medium-size ship like cruisers, although destroyers or t1 exploration frigates are great at the role too.
For mining you have a few specialised cloaky mining frigates that can do the job well, but have lesser yield than the barges, which are medium-sized ships.
For hauling you’re very limited in volume if you stick to frigates. For low volume high value items you could use the frigate-class Interceptors to quickly haul items, but for most hauling jobs you will need industrials, which are medium-sized ships, or freighters, which are capital-sized.
You can stick to flying only frigates and you can do a lot if you only fly frigates. Especially as new player it’s nice to have a training focus (there are just too many interesting skills to train otherwise 
) but don’t let it limit you: many other ship classes are easily trained to a basic level if you need them.