But I am not sure what champion would potentially be the easiest to start out with. Also if you have a link to any good guides for jungling on the new rift I would greatly appreciate them.
Jungle champions I own: 
and maybe
?
I recommend buying 
or 
.
They are very cheap, have decent clears that are reasonably safe (very safe for nunu), not-terrible ganks pre-6 and GREAT ganks afterward.
If you're just learning the jungle and want to get down gank mechanics, timers, pressure ect. these two are pretty great.
Master Yi is the easiest one of those champs to learn. Try him.
Nid and Shaco are really mechanically challenging if you want to get everything out of the champs (learn them once you got the basics). Diana and Malph can jungle, but they aren't good solo queue champions since they can't really pressure lanes with ganks until they get their ults at level 6. Eve could work, but she's tricky to play well (need to be in the enemy jungler's head) Fiora is a no. She offers nothing to help your team out of the jungle and is easily countered. Your team will do nothing but rage at you and you won't have a lot of fun jungling.
My $0.02!
I first started jungling as and then , and right now my go-to jungler is 
.
I'm Bronze I so I'm no expert but I've jungled since I started playing the game. Jungling is more about objectives and getting your team in a healthy spot which can be hard, especially in solo-queue where everyone wants to carry. If you don't mind taking the blame for most mistakes in the game and can multitask then jungling is really fun. I actually want to start playing Diana again
Nid is strong, but hard to play. Malph is kinda situational. Yi and fiora can be strong if you're the type who plays that kind of champ in lane, personally I'm not big on the farm heavy jungle who kinda needs to snowball a little.
My advice would be to look at buying ww, nunu, udyr, xin, or amumu depending on the play style you think you would like best and what runes you have.
Personally I love nunu utility and he is really safe in the jungle but he isn't everyone's style. Check out stonewall008 on YouTube for some videos that can give you a bit of the mindset you want as jungle and can let you look at some of the cheaper ip jungles to see if they look fun to you.
Do yourself a favor and refund Yi. You'll have more self-respect when you do so.
all of those champs can jungle but a champ that HAS to farm alot to become effective is less effective overall than a champion who can go and own any time during the game. right now from your list, i would say diana is the strongest, easiest and most efficient one. though her early ganks being weak, she starts snowballing automatically from level 6-8 if you didnt get raped 5 times. she is very tough and its really just worth to learn her.
Trundle or olaf are the easiest clears you're going to get. If you want to work on ganking try amumu and his clears get decent after the first four camps or so. Once you get a grasp on each of those bring them both together. udyr can farm and gank pretty well. Also go into custom games and work on your clearing. you won't always get a leash so it's good to have an idea of how to do it solo. Once you can clear well enough in a solo game add some bots. Keep going until you're comfortable with it.
And to be quite honest once you get a trailblazer the jungle is the easiest thing in the game. farm until you see something worth ganking then either get a tower/dragon then back to farming.
Here's something you're going to have to learn if you want to jungle and I've had to learn it the hard way: You're not a super hero. Be opportunistic as hell. If you can't gain something from ganking a lane don't bother with it. You have one job no matter what anyone says. Your job is to get towers down and win the game. Rushing in like superman to help the buffet lane is only going to get you killed. You can show your self near a lane to relieve some pressure but you dying is only going to make things worse. And lastly ALWAYS hit tab before you gank. If you're thinking of running in to help your top laner who is against the 5-0 rumble just don't go near it. Turn around and see if you can do something elsewhere. They will get pissed but there is absolutely nothing you can do there without bringing the rest of the team. Your job is to win the game, not someones lane.
Nunu is a great jungler to learn on right now, and extremely powerful objective-wise.
I think you added the bit about links to guides. Seriously, when I jungle mained a few seasons back these youtube channels were what I watched whenever I was bored. You get great exposure to different champs you might enjoy, and their commentary about why you do stuff is usually really good.
https://www.youtube.com/user/stonewall008
https://www.youtube.com/user/foxdropLoL
Also nightblue3 used to be pretty informative on his stream but I don't know about now. I generally watch phasekiller streams if I'm watching streams now.
It doesn't get simpler than 
. His ganks aren't that great though, regardless of build, so you may want to try someone with a better gapcloser (, 
, 
). If you want the carry-est of junglers, it's probably .
If you're willing to drop a little more IP, 
is currently the best tank jungler and has a fairly simple kit with a lot of power. I also think 
is the quintessential jungle champion, but she has a tricky kit if you're not used to it; it can take some time to master. Plus, she's not a 'strong' jungler right now, per se.
is also a really standard AD bruiser jungle champ, so it's good to know her.
So yeah, there's a few options. All of these can clear well without much difficulty, too, so you should have a safe time in the jungle.
This is the tank meta season so tanky junglers are the preferred flavor of the month right now: 
but Nid and Yi are always good picks. Personally when i started jungling i first began with Sej and Amumu--Sej was always incredibly good, not just now.
As for guides you can find plenty on mobafire--a good website for getting your bearings with new roles and champions, but im sure a high tier jungler will just reply to this topic with what to do and how to do it.
