Which weapons you can use depends on your level.

You can tell if you can use a weapon by looking at it (use 'see'
to look at weapons in the shop)

For the first 4 or so levels, the free weapon you get at logon is
likely also the best weapon. At level 5, there are some good alternatives
however.

A short list of some of the popular weapons and their min usage level:

zip gun : level 4 (blaster)
A Subsonic THX-0138 Junior Cannon : level 5 (blaster)
Ancient dagger : level 5 (melee)
Shiny microblaster : level 8 (blaster)
Standard lightsaber : level 10 (saber)
Miniblaster (improved) : level 11 (blaster)
Maximator : level 11 (blaster)

Note that while you can use a lightsaber at level 10, it is heavily
skill dependent, and is not guaranteed to be a good weapon unless your
force skills are very good. Also note that it will not auto activate
untill level 15

Also note that the 'free' weapon trains your skills, but does not depend
on them much, unlike all other weapons.