Most likely you haven't pushed it down far enough on the valve...it can be quite a snug fit, even with the lever down. You might be able to screw the outer ring of the pump head tighter or looser to adjust the fit.

Schrader valve is like a car tire valve. Unscrew the black cap and you see a round end with the valve post in the centre. If your pump head has two sizes of sockets, fit the larger one over the valve, press it down, and lift the lever.

Presta Valve is skinnier, and after you remove the cap, you must also unscrew the end part of the valve. use the smaller socket on the pump head, and push it as far on the valve as possible, then lift the lever.

Some pump heads have one socket, but removable parts that can be flipped over to accomodate one or other valve type.

It's normal to hear a little bit of air escaping as you detach the pump head. It probably mostly comes from the pump not the tire, so it doesn't mean your work was wasted.

A lot of mini pumps have difficulty pumping much above 60 psi. For higher pressures you may need a floor pump or frame pump.

Be careful when you pull the pump off the stem to pull straight and steadily. If you wrench it from side to side you may hurt the base of the stem at the metal rim of the tire.