I have 3 problems with my maytag washer (don't know if the problems are linked). The first is that the pump is leaking. 2nd is that when it is on the spin cycle, water isn't pumped out before it starts spinning and it puts water everywhere (probably linked to the pump leaking) but eventually works properly. The 3rd problem is when the machine goes through it's cycle, whether it is washing or spinning, it doesn't want to work at full speed - I push (stop) the dial and start it again and then it seems to work at the proper speed.

Hi from retired Englishman in SW France,

Providing your particular model has one I would gently suggest as a first step that you clean the drain motor/pump lint tray/filter- so many 'faults' are caused by blocked filters.
Following are my complete and hopefully 'coverall' pre-typed guidance notes for checking both the drain motor filter and the drain motor ;-0)
REMOVE POWER AND ENSURE YOU HAVE A WATER RECEPTACLE/RAGS TO SOAK UP THE WATER WHICH IS BOUND TO POUR OUT (alternatively use a wet vac as sensibly suggested by VinnyB1234!)
If you are unsure where the filter is, it is often behind the front lower (kick) panel (or in some machines at the rear in an awkward position!). These panels are either held on by (plasic) screws or will just pull off. Most filters are held in by a round cover/knob. Sometimes the filter will not come out with the cover and you will have to persuade it to come out to play.
It's surprising what people have found in theirs! It is always worthwhile checking it on a regular basis.
Now- whilst the filter is removed it is normally possible to see the drain motor and ensure it is not jammed by a foreign object such as a needle or a toothpick. It should be able to be turned by hand/finger- the power IS OFF!
At this point it is also possible to test if the motor works. No clothes or water in the machine, select a spin cycle and put the power back on. You should be able to see the motor turning at some point, possibly straight away. Using something suitable like a glove see if it is easy to briefly slow the motor down. If it is easy then the motor is malfunctioning.

If this has assisted you please consider a 4 thumbs up for the rating.
Thanks and good luck!
John C