Almost always start by your bottom lane, whichever mini camp is closest, or buff camp in certain scenarios. Pantheon on blue side, for example, wants to start red as it helps him sustain in the jungle 10x more than krugs will.
Amumu is fairly trash tier for jungling as far as early pressure, he has zero - but does extremely well in team fights in terms of hard CC and engage, similarly to Leona. Miss your Bandage Toss, waste your time and give away your position.
Kayle you're generally looking to start at a buff camp, it takes a little bit to get used to her clear as for the first few levels you'll find you burn through mana - a lot, and she's one of the few AP scaling jungle champs you don't build Magus on due to her E being AA based.
Maokai I have no experience to speak on, but he does make for an annoying time to try to escape a gank.
Nasus scales his Q relatively slowly in the jungle and has close to no early presence (lvl 2-4) where a lot of the current junglers shine
, his base damage is fairly low and you can usually duel with him effectively early.
Riven has slower clears now than S4 and takes a fair amount of damage given her relatively long E CD early on and short length of the shield, usually require a hard leash if you want to be able to gank off your first clear.
Warwick has great sustain, little early presence without a gap close, but at 6 he makes a lot of shit happen.
Voli's just great all-around, no argument there to be made there.