You don't need to have the entire $10k. If you buy the 'Investor shares' version of the Vanguard fund you are interested in, the minimum to open/buy is usually $3000. If you continue to add to your account, they will automatically convert your shares to the Admiral class once your balance has reached $10,000. It's what I did with three different funds in my IRA. It is also significantly simpler than dicking around with Wealthfront and dealing with whatever fees they charge to close out your account once you have enough money.
Since I'm assuming Wealthfront doesn't put all your money in a single fund, you would probably have to deal with capital gains since you'll be selling something. (Even if they let you transfer shares, which I don't think is a thing, but I could be wrong).
The real thing I would do is read more about investing (Boggle heads forum, that stock series that everyone recommends). Then, I would compare the expense ratios of the Vanguard fund I am interested in vs what would happen with Wealthfront (I have no idea how their system works). I would also scour the Wealthfront site for info about fees for closing the account and whatever else they charge you to make money. Then I would use that information to decide.
Personally, I think you'd be best off getting the initial $3k together (using a basic savings account till you've got it) and just starting with Vanguard, but I haven't done any research on Wealthfront.