As for counts, I really can't say cause it varies from champion to champion.
I will say the best advice I can give is the following guidelines.
IMPORTANT THING TO KNOW
Before I can't tell you anything you must understand your ultimate goal, its not just to take the nexus. Its to control the map! Think of it as an RTS, in an RTS's the player with the most map control generally has the most power and will most likely win. THIS Goes double for games like League of Legends. At the beginning, the map is completely even, and the goal is to make it uneven. Constrict their available movement. How? By taking towers and warding! This will give you more influence in the map. For example:
You take all the enemies first towers, Now your line that you can move has moved up, in lanes tho, they could still be in the jungle. So because you warded the jungle you can see where they are. Now they have no safe zones past their tier 2 towers, cause you express your control over the warded jungle zones and the lanes. Hopefully I'm making sense. Moving on.
PRESSURE
One of your top priorities as a jungler is plays off what the other team is doing. If you are in bottom jungle and their jungler is top trying to gank your top. Get drag (high priority), gank another lane, or counter jungle(steal enemy camps). Most of the time, you waste time trying to run top and help them when you could be pressuring another sector of the map.
Something very important to note here is that, contrary to what some people say, taking a tower quickly is good! If you successfully ganked top(forcing him to recall or better, die) and you see the enemy jungler bottom, TAKE THAT TOWER! This gives your team an early advantage.
WARD
This is by far the most important!
The best way to counter play the other jungler & team is to know where they are so you can react quickly. My personal rule is 3 ward in the enemy top jungle and 3 wards in the enemy bot jungle. Coordinate with your team or someone to help you with this since you can only place 3 wards; AND PLEASE LEARN THE BEST PROPER WARDING SPOTS! :D I'm sure you will run into a lot of games where people put wards in spots that show little to no vision.* The goal is to maximize vision, not just to see in bushes*!
TIMERS
Thanks to Riot the timers are now on your tab, but only if you see it die, so keep an eye on it with... you guessed it.. WARDS!!! :P
Add me and I'll be happy to help and play with you some!