Betterment charges 0.25% of assets, but you can manage things yourself and save that 0.25%. Just keep Vanguard, Fidelity and Schwab in mind when you're deciding where to park your investments.
I think Fidelity now allows individuals to open/transfer a health savings account (HSA). So you might be able to have your Roth IRA, HSA, and taxable investments all in the same place. If you're new to investing, though, Vanguard is better since it's much harder to make a mistake.
Are you willing to delay your home purchase if the market corrects? Meaning, if the market drops significantly, your down payment might no longer be enough. If you can accept that risk, and wait 1-2 years (based on most corrections - not guaranteed) after a stock market correction, then you might be able to invest your down payment in stocks and bonds.