Better tyres for Cannondale Bad Boy?
You've really got a few things going here. 

For going up hills easier, you have 2 aspects for a given bike:


Bike geometry and bike fit: If the geometry of the bike is not good for climbing and/or you're poorly fit to it, its going to be harder than necessary. So play around with adjusting your saddle height + position + stem + handlebars.
Gearing: For going up a hill, you want a low gear (big cog in the back, smaller chainrings in the front). Your tire size does play into this a bit, but its secondary (bigger tires = higher gearing). You can use something like this to calculate gearing. 


So, if you want to make it easier for a given bike to go up a hill, fit a cassette/freewheel with bigger gears, and use smaller chainrings in the front. Note that lower gearing means that your top speed won't be that high.

When descending a hill, you also have two things:
- Bike geometry and bike fit: The position of you on the bike and the geometry will determine the stability and control you have when descending.
- Brakes: Brakes are the way you'll primarily stop. If its a big descent, you have to be careful not to overheat the brakes/rims, so you need to choose appropriate brakes (this includes pads, calipers, etc.) that provide the right level of braking for the weather/terrain and modulate them appropriately (e.g. if you go down a long hill and just hold the brake partially on, you may boil your brake fluid on a hydraulic brake, just like you would on a car). 

Your gearing won't necessarily matter all that much on a bike with a freewheeling mechanism, since you'll either have a gear that you can accelerate with or you'll be freewheeling (and if you want to accelerate, you'll want to add combinations of higher gears (large chainring, small cog in the back)). 

Tires: For road use (on a clean/wet road), you want smooth tires. Knobbies increase resistance on a road, plus when cornering at high speed can flex and lead to wiping out. Also set your tire pressures correctly (this is important for comfort and control). Off-road or snow will favor some form of knobbies. As for tire sizing, note that it does affect the standover height a bit, but this is a personal preference determined by how much you weigh, how much comfort you want (bigger tires at lower pressure are more comfortable), how well the tire stands up to cornering (which is determined by how flexible the tire is as well as the pressures its run and its size -- a big low pressure tire may flop off at a high speed corner when a thin high pressure tire wouldn't), puncture protection, rolling resistance at a given inflation pressure among other things. Note that you don't want to run a very wide tire on a very narrow rim and vice versa due to the tire coming off or hazards damaging the rim. 

Unless you're riding on a road and you have knobby tires, I'd probably start with finding appropriate gearing (by swapping freewheels/cassettes so you have a big enough big cogs for the uphills and small enough small cogs for the downhills) and making sure brakes are up to snuff (probably starting with having good brake pads and the brakes are in adjustment). If you are riding on a road and have knobby tires, buy some slicks.