It sounds like you just need to get a feel for traction. Although bike geometry and tires will affect the traction threshold, the main thing is that you need to ride within the limits of your bike.
Keep in mind that managing traction on a bike isn't at all like a car. With a car, your body is a tiny part of the mass of the vehicle so doesn't play a major role in vehicle handling. On a bike, your body's mass is much higher than the bike itself, and your feet are directly driving the rear wheel, so your body's position, movements and your pedalling technique are all important.
A few factors affect traction on the rear wheel. Tires and riding surface play a role by increasing or decreasing the threshold at which you lose traction. The amount of your weight pushing down on the wheel also affects traction. This depends on your position on the bike and the slope. If you're going uphill out of the seat with your weight forward, for example, this unweights the rear wheel significantly. Acceleration and braking forces also play a role. If you brake hard with the rear wheel or accelerate hard, you transmit a lot of force through the tire to the ground. When accelerating in poor traction conditions, you want to make sure that your pedal stroke is as smooth as possible to avoid spikes in pedal force. Cornering also plays a role, producing lateral forces.
You really need to get a feel for how much traction your bike gives you on different surfaces: then you know how hard you can corner/accelerate/brake without losing grip. You can improve this with technique: for example by moving your weight back to increase weight on your rear wheel, or by easing up on braking or acceleration through tight corners.