There are several useful observatories you can build on the Moon
Infrared observatories in the Lunar polar craters of eternal night - the coldest spots in the inner solar system, permanent darkness, passively cooled
Radio telescopes on far side of the Moon. Moon provides shielding from interference from radio transmissions from Earth. On the Earth we can only observe the radio sky in a few selected radio bands set aside for astronomy - and even for those - you get interference from some stray signals.
Daedalus crater on far side of Moon especially useful here>
Using the lunar gravity and a spinning mirror - and with natural craters already roughly parabolic in shape - the Moon is an easy place to build very large liquid mirror telescopes - reflective liquids forming a natural parabola shape when spun in a gravity field
We have those on the Earth - but none in space - and makes an easy way to create really huge telescopes without need to send a huge mirror into space from Earth.
All these ideas would need more research of course.
For far side radio observatory - simplest long range telescopes consist of no more than wires spread out over the surface (like the first radio telescopes on the Earth), and would be easy to construct - possibly using telerobots controlled from a mission with humans in the L2 position.