Right now, if your Verizon handset is LTE capable and you want to stick with Verizon, your only option is Page Plus/Straight Talk, both owned by Carlos Slim's America Movil. PP used to have reasonably good domestic customer support, but that will be ending in January 2015 as the call center gets borged into the general AM calling centers... which can be
unpleasant to deal with.
Now, if you do have a Verizon LTE handset and it's got full US GSM band support (you'll need to give me the actual model number or look it up yourself), it should be carrier unlocked and if AT&T coverage is acceptable, that'll open up a whole mess of GSM MVNO options. AT&T GSM coverage is about on par with Verizon coverage (excuse some more desolate parts of Nebraska) as they're the two 800lb gorillas. You'll also have more phone flexibility, cheaper handset options, and cheaper calling package options going GSM (if it's an option with your current handset or with coverage in the areas you frequent, or in the future when it comes time to replace).
You most likely could also replace the battery, and there's a chance the handset is supported by
CyanogenMod, which might permit you further and newer Android updates beyond the official update releases extending handset life that much further before obsolescence or hardware failure overtakes it.
As for other Verizon MVNO options, there's not many who have had their mettle tested as of yet that I'm keeping my eye on. There's
Selectel,
Red Pocket's new CDMA Verizon plans, and Page Plus Master Dealer Kitty Wireless' new
Puppy Wireless venture (I've recommended Kitty as the preferred dealer for PP users for years), but none will have LTE support until at least Q1 '15. If your handset is a non-LTE CDMA device, you should be able to activate now without any problem, but if it is LTE and you want to go with one of them, you'll have to wait until LTE support is rolled out with them before you can switch. Selectel has been on the market long enough at this point that they're a marginally safe call now (they're past the hump and pushing two years), but they're still not anywhere near the established camp that
the headline MVNOs of the guide are. Puppy is still
really new as is Red Pocket's CDMA service, and unless Verizon has changed their MVNO policies (which is what did in TalkForGood with Big Red back in '13 - and countless other MVNOs over the years, and they haven't as far as I understand - but for all that I know, I know there's more out there that I don't), either could be a gamble for longer term service. That said, they're still both established and reputable companies... but so was TFG. Hypothetically, if either of the two newest Verizon MVNOs who were to fold up their service, I would trust Kitty to be the most likely to notify customers of a pending shutdown to permit their customers to port out.
That should get you started. Any other questions, give the guide a read, and don't be shy asking.