General starting jungle advice:
Regardless of your champion, make sure you can clear your camps competently. Practice in custom games. There are times where you won't be able to gank any lanes, and you should still be making yourself useful by getting more gold in that time. At the very least, make sure you're able to do either one full jungle clear, or get to the point where you go back to buy your first item (on champs that can't do a full clear). You're never guaranteed to get a leash in game, so you should be able to do it on your own when you need to.
Counterjungling / invading is fairly advanced. It's high risk, ??? reward. You need to have a good idea of what's important in the jungle through personal experience (and probably getting counterjungled) before you should try it yourself. Because even if you don't get caught, a lot of the time the camp you kill will respawn before the enemy jungler even sees it, or you'll spend so much time running through their jungle that the enemy jungler has ganked twice in the time it took you to steal wolves.
Champ choice:
For learning, you should make sure you have a jungler that can do somewhat reasonable ganks at level 3. (old Warwick, Nocturne, Shyvana, and Master Yi are all champs that have no CC, no gap closer, or are missing both, until level 6.) You said you want to make an early impact, so that's a good fit, lol.
Kills, early power, and setting things up for your team sounds like Lee Sin, Xin Zhao, Vi, and Elise. Shaco can get kills in lots of ways, but he lacks the CC to do much setup (until/unless you are veeeeeery good with your jack-in-the-boxes).
Depending on how weak you are to teammates calling for specific team comps, I'd also recommend learning a jungle tank (read: Amumu/Sejuani/Gragas/Zac). They play fairly similarly, and they all can pull off decent ganks at level 3. They then also have great disruption influence when they hit 6, similar to Ori and Zyra that you said you play. None of them are going to be getting the kills themselves. But Zac especially has healthy clears and very strong early ganks. (I mean, when he's not meta, that's the reason you pick Zac: he's the tank jungler with extreme range on his ganks.)
Edit: But of course, the most important part is having fun. So as long as you choose a remotely meta jungler (like Shaco) and stick to it, you'll learn a ton!
Edit2: I guess I should look at the newly updated Warwick. Changing references to mention "old".