I transferred 2 accounts out of Morgan Stanley - one to Vanguard and another to Fidelity.

I don't know how Wealthfront and Betterment work, but from Fidelity and Vanguard, you can initiate an account transfer from their website.

Whoever was assigned to your account from Morgan Stanley may call you or email you to confirm the transfer and stuff, but it's more or less straight forward.

In terms of fees:

You will likely incur an account closure fee from Morgan Stanley. I think it's $75 for most accounts.

In terms of other fees:

Make sure to transfer your securities IN-KIND. In-Kind means they will NOT sell your stuff before they transfer it. Your investments will just appear in your new accounts. The only thing you should do is record your investments (like names, # of shares, and cost basis) in the unlikely chance it gets jumbled up in the transfer.

Cost basis will likely not transfer over for over a week. And the process itself can take up to 2 weeks. So, don't transfer anything that you need to touch in that time frame.