I recently moved across the country (about 3 months ago) and was wondering if I am able to sell any of my equity without a huge tax penalty. I definitely acquired the stocks long before I moved (more than a year ago), but I am ready to sell them now. I want to ensure that I wouldn't need to report the capital gains to both states and then pay long-term capital gains tax in both states (both states have state income tax).
It depends on the tax laws in the two states. Most states either tax only income that was earned while a resident of that state, or tax proportional to the amount of time you lived in that state. So you might pay slightly more or slightly less tax than if you were in one state only.
If you use tax software, you could try filling out "fake" last-year returns for each state including capital gains and see how different the results are. Or talk to a CPA in your current state to see how your gains will be taxed.