Can I run wire from my meter to a shed for a subpanel?
No, the cabling run from service to shed needs to have overcurrent protection.   Your local authority might allow it under the tap rules, given a pretty good reason... But "out of space in my panel" definitely won't qualify.  

It sounds like you should start with a subpanel in your home to move the smaller loads out of your main panel.  This is a big job like the Platte river is wide (but not very deep).  And it's very DIYable at sane cost. It will set you back a panel (some of which come with free breakers), a big double breaker, and some breakers if your new panel uses a different type (which may be a good idea if your current panel is a builder-grade cheapie, or if GFCI and AFCI breakers are unduly expensive for your panel.)  And of course wire and the odd junction box or bit of conduit.  

If your existing cables are too short to reach the new subpanel, install new junction boxes to where they can reach.  Splice them there onto some more cable to the new box. Bonus points if you can make the box-to-panel run in conduit; but only 4 circuits are allowed per conduit.

Get the biggest panel you can fit; a few extra spaces are cheap, and often even come with more free breakers.  60 spaces is not too large. 

Leave your main panel feeding only large loads and subpanels.