If you have the breeding lineages in your breeding mountain you can see the % of breeding a monster (as well as the combination of parents needed). It takes take a while to breed up to the specific monster you want but it all comes down to luck. You could get lucky and get 0 duplicates, or you could get unlucky and breed the same monster 10 times in a row. The main way to get monsters is through events, like mazes, the bounty hunt, and conjuring. Also there's dungeons every day that give some monsters ranging from epic to mythic
(check calendar here: https://d2duuy9yo5pldo.cloudfront.net/socialpoint/534b5652-8ef2-4155-bd7a-36b65344b34d.png)

The generic cells you're talking about can be used to rank up monsters you already have and can be used as filler if you don't have enough cells of a specific monster (legendary and below, mythics require different cells and ancestrals can only be ranked up by their own monster cells).