This is going to be a somewhat short look at the subject because there are a lot of factors involved and it is going to be from a defensive perceptive as I don't have a lot of experience with offensive invades.

Getting mass invaded by 4 or more of the enemy team is a separate topic. Usually look and see how much CC the enemy team has or if they have a Thresh, Braum, or Blitzcrank, to estimate your chances of a mass pre-minion invade on your jungle.

It is hard to count on your team mates in the early laning phase to deal with invades by the enemy jungler. They tend to be more focused on last hitting and setting up how their lanes are going to play out. (Though if their laners are pushing like mad as soon as they get to lane then it could be a tip that your are going to be invaded). Having your own laners stuck under their turret trying to scrap up what farm they can while under pressure will generally prevent your laners from responding to an invade by the enemy jungler.

The worst scenario I tend to run into with respect to invades is against someone like Graves or Shaco. If they get a kill on you the better ones will keep hounding you for the rest of the game and using the interludes when you are dead or off on the other side of the map to gank your laners from out of your own jungle.

Once they have their lanes ahead you can often kiss at least one side of your jungle goodbye for a good chunk of the game and the heavy lane pressure that they are able to then apply means that is usually unsafe to try to make up the difference by going into their jungle.

Total yuck.

Part of avoiding invades is simply experience. You can predict to some extent if your are going to get invaded at champion select. Shaco, Graves, Lee Sin, Kha'Zix, Rengar are all candidates that like to invade junglers that they feel they can out duel. However you can't always rely on that. I've had a wild Zac show up to screw me up before. * Ivern is a different creature. He won't try to out duel you so you have to deal with him differently.

They tend to be champions that can either clear their first buff super quickly, or cases where they get a heroic leash from their laners - Malzahar and Zyra give boss leashes, or they are very good duelests that have kill potential at level 2. So they end up so healthy after clearing their first camp that they feel comfortable trying an invade. That happened to me in the case of a Warwick. He got a hero leash from a Zyra on Blue Buff, ran to his Red Buff, use the blasting plant to get over the Baron pit and invaded me at my Blue buff right after I had smited it to try and get some health back, because I wasn't expecting an invade from Warwick who tends to clear healthy but relatively slowly (I think I was playing Hecarim at the time and started Red Buff > Wolves > Blue, which leaves you very low on mana and rather low on health when running just a Refillable Potion and Hunter's Talisman for sustain.

So your first step to avoid being invaded after champion select is to try and get a ward where you think the enemy jungler is going to start (which is often the same side as their Bot lane because they can get a leash from too champions as opposed to one), which will tend to be Blue or Red Buff.

With that information you can decide your game plan. However, if you can't get a ward in safely or no one shows up on your wards you are just going to have to play it as best you can.

So you are Blue side. You get a leash from your Bot laners on Red. Your intended route is Red > Wolves > Blue for level 3 and double buffs > look to gank.

After Wolves use the blasting plant to hop over the wall behind the Blue buff encampment. Aggro Blue and drag it into the bush and proceed to take it out.

Using the blasting plant to get over the wall prevents the enemy jungle from making use of it for the same purpose or for making a get away after a steal.

You also have the option of warding assuming your trinket is off of cool down at that point.

You can also decided that instead of aggroing Blue into the bush you can pull it to one side so that your back is against a wall that you can Flash over.

Lee Sin's like to throw a Q over the wall to see if the buff is still there, so having it in the bush or off center will sometimes result in the Q missing which can throw them off their game plan.

If you do end up with a nasty jungler at your throat and you don't think you can win the fight, then if you have it you want to drop a ward, and get to safty as quickly and as loss free as possible. If positioned well the ward will give both you and your laners intelligence on what is going on and help you and them make decisions about how to react.

If you are on Blue side and going Blue > Wolves > Red you can take the same approach though you have to go a bit out of your way to take out the blasting plant (if you aren't planning on using it to go over the Dragon pit wall after finishing your Red, which you should only do if you are okay with meeting the enemy jungler in the river).

Anyway, you can spend quite some time thinking about these sorts of scenarios. You'll get better with it over time, especially if you watch your replays and see how the invade went down.

* Iverns tend to purposely start on their opposite buff from you and then use their passive to setup two camps for later dismissal and then will immediately run to your Buff which they will Smite and their isn't really anything you can do about it other than scout out where they are going to start and go to their opposite Buff and start your jungle there (hopefully with the help of your laners on that side of the map). Also ask your laner on that side of the map to drop a late ward at your buff so you can know if Ivern invades. This is worth your laners time because many Iverns like to level 2 gank from behind, which can be deadly.

You can try and start as normal and when your see Ivern on the ward by your opposite buff you can run to his Buff of the same type and take it but this is at the risk of being collapsed on by his laners if they become aware of what your are doing since no other jungler can virtually instantly clear a jungle camp with only Smite.