Can you use and move around a telescope on Shabbos?
Yes, if you hold that you may study science on Shabbos, the Shulchan Aruch says, in Orach Chaim 307:17:
ויש מי שמתיר ועל פי סברתו מותר להביט באצטרלוב בשבת
that there are those who permit studying "sifrei chochmos" (secular studies) on Shabbos, and that according to that reasoning, you may peer into an astrolabe on Shabbos.
Use of binoculars, a telescope, a tripod and glasses is permitted provided that there is no other prohibition involved such as carrying in a public domain, use of electricity, and on the condition that special skills are not required to assemble the equipment. It is permitted to adjust binoculars and a telescope in the normal manner. (Shmiras Shabbos Kehilchoso 16:45)
However, see footnote 111 there, where Rav Neuwirth brings an opinion that a telescope may be muktzah machmas chesron kis.