The problem with cell phone voicemail is that it's controlled by your cell carrier instead of you. Some carriers delete messages after a certain amount of time; even if you want to save that one special voicemail, you're out of luck. As far as I know, the big carriers don't let you use different voicemail greetings depending on who's calling. They don't email text transcriptions of your voicemails to you, either.
The solution: YouMail.
YouMail gives control of your voicemail back to you. Set custom greetings for different callers, get transcriptions emailed to you, save voicemails for as long as you want and much more. You just follow the instructions to configure your cell phone to forward calls to YouMail instead of your carrier's voicemail system and then you tell YouMail how to handle those calls.
The free account is ad supported and doesn't offer voicemail to email transcription (although I think it does give you a limited time trial of the feature for free). There's a paid version that's still pretty affordable (under $20/year) and gets rid of the ads. I haven't really looked into the paid version(s) because the free one is good enough for me.
Check it out: