You should always start at the bottom lane, so it depends on which side of the map you´re playing. Start at the Golems or the Gromp and smite it instantly. Get a nice pull from your botlane and go ahead for the nearest buff (red or blue). Clear it up. Walk to the next buff (red or blue) and clear it up with using smite. If you´re getting low hp at redbuff you should maybe smite it earlier to get a heal out of it. Be careful with counterjunling, and maybe put your trinket-ward down at the 2nd buff that you´re taking. After that go for a gank if you have enough HP or go back, and just keep it going.

Ganks are really easy as an Eve. You don´t really have to be scared of wards, because you´re invisible if you walk in for a gank, that´s why Eve has a very strong and useful early ganking advantage. You should use your smite always, if you´re in range of autoattacks for using the whole damage from it. In early (LVL 1-5 or without phage) your ganks are weak without a redbuff, so you should do some nice earlyganks right after your first jungle route, that you´ve taken, and just farm up until LVL 6 or until you got your phage for the next ganks. Of course you can also gank without it, but you maybe won´t be efficent, if the laner hasn´t any CC for helping you out chasing.

These are very useful, just like everyone knows. You should always try to do some nice and safe calls. If your team is behind, these can get you back in game, but can also loose you the game, if you get of a bad one, so you have to be really careful. If enemys are doing it and you´re sure that no enemy pink is there, you can walk in and try to steal it with your smite. But guys, just like always, safety first. If you think it´s too risky, just don´t do it. If you think its pretty safe and you´re good at smiting or the enemy smite isn´t up, just go for it and try your best!

Here it depends on your Item-Build how you have to act.
If you´re tanky and ahead of the game, just go ahead and engage with ur R and W and use your Randuins for your team to follow up. If you´re in there, you should be able to deal some nice damage to their carry, but if you´re getting CC´d too hard, wait for your team to follow up and get yourself some peel with walking behind them.
If you´re based on damage or behind of the game, wait for you tank in team to engage, or the enemy team, and go in sneaky to reach their carrys. It´s really good if the enemy team is engaging, because the ADC for example won´t get too much peel then, so you can burst him down easily and if you got 1 or 2 of their carrys, your team should be able to finish the rest of the enemy team.