People around here have mixed feelings about Betterment. A
three fund portfolio will work just fine for most investors, and this should not be difficult to build yourself without paying Betterment a management fee.
I recommend not starting with a taxable account. Those are usually best for people who are already contributing the maximum to their IRA (either Roth or traditional) and any workplace retirement plans.
As to whether you should go with Roth or traditional, the basic question is whether you expect your tax bracket to be higher in retirement than it is now. If you expect a higher bracket, go with Roth; if you expect a lower bracket, go with traditional; if you expect it to stay the same, pick either one. Most Mustachian early retirees will likely have a lower income when they retire, so traditional is often preferred. Given your low current income, you may actually find Roth to be better in your situation, especially if your teaching job is likely to pay a decent pension when you retire.
Do you have any tax-advantaged savings options available through your employer? Look for numbers like 401(k), 403(b), or 457. You might want to start there rather than opening your own IRA. The contribution limits for these are often higher, and many employers match contributions.
As to what you should do with your $10k, investing all at once is likely to be better than investing a little bit each month. There are no guarantees, but the odds are in your favor. If you do decide to go with a workplace retirement plan, those have contributions through payroll deductions only. Set your contribution to be most of your paycheck and spend the money out of your savings account. This is effectively "transferring" that money to your retirement account.