I have an expired and suspended account at Servage.net and I want to delete the account and then re-register the domain somewhere else at a better company. Would the domain be available for registration after I cancel my account or will it still be in the system or something? (I'm rather inexperienced with this stuff...) Is this a bad method? I simply want my domain to be registered somewhere else.

This is a bad method.
The reason is because once it drops and become deleted from the registry, anyone can register it.
Even if your domain is really odd and weird, there's at minimum a 60% chance someone will pick it up, ie domain squatters.
My advice is to renew the domain for 1 year at servage, then initiate a transfer from the registrar you want the domain with.
Edit: After reading your post again, sorry:
Yes the domain will still be registered until the expiration date shown in whois.
You will have to transfer the domain and ask servage for EPP code.

Delete account and then register elsewhere?
Only if you can afford losing this domain for good.