Slipping is where the clutch and pressure plate are making contact but not fully engaged. This happens every time you take off and to a smaller degree when you shift, this is a normal part of the process. However, some people use it as a way to be lazy on a hill to hold them in place instead of using the brakes. It is possible to engage the drive line enough that it will hold you in place on a hill but not actually start to move forward. This will ruin a clutch very quickly. When a clutch wears out it will start to slip when fully engaged. When this happens you can tell because you hit the gas and instead of it reacting evenly with the amount of gas you give it, it accelerates much slower, until it stops engaging altogether. With a slipping clutch you can be rolling in gear, gas it and watch the RPMs go up without it accelerating at an equal pace.
With a tach that doesn't work correctly you are going to have a difficult time. I learned on vehicles that didn't have a tach in the first place which meant getting a feel for what the engine was doing, which unfortunately for you is mostly by hearing it. I'd suggest getting that tach fixed.
As far as what to do when shifting - the taking off part as already been covered by others. Give it a bit of gas (not too much) then when you feel the clutch engage start giving it more gas while lifting off the clutch. Between gears, always lift off the gas, otherwise the RPM of the clutch will be way higher than the pressure plate when you go to release the clutch. This means you will either get slip (bad) or with a grippy racing clutch an unpleasant jerk. Once you get comfortable with driving a manual, look into rev matching for downshifting. It makes downshifts much smoother which makes the ride more pleasant and is easier on the clutch, but that's not something to worry about until you have the basics down.
Practice on flat ground quickly transitioning from brakes to gas and timing the releasing of the clutch for takeoff. If you can get good at that without stalling then hill starts will become easier to practice when you feel ready. The only difference is you will need to give it a bit more gas so you don't stall out when on a hill.
Also, always remember to push in the clutch when coming to a stop, or you will stall.