Rollover IRA for the 2 old accounts.
Remember that the 401k max is pretty low. Save more.
Don't use a rollover IRA because it limits your backdoor Roth in the future.
If your new employers plan allows it, and most do, roll both your 403bs into your current 401k. Pick some combination of broad, low cost mutual funds or ETFs (we can help you decide which, but it depends on which are available). After that, your saving priorities are, in order of importance:
1. Health savings account (if offered given your current employer and health insurance) $3450 max if single, $6900 if married
2. Max your 401k and any tax advantaged accounts available to your spouse (if applicable). This is a limit of $18k per account, where each person can have 1-2 (second being a 457 if applicable)
2. Backdoor Roth IRA for you and your spouse (if applicable). This cannot be done easily if you have any old pretax IRAs around. Maximum here is $5500 per person.
3. Mega-backdoor Roth IRA if your employer 401k allows both after tax contributions above the cap AND in-service rollovers of after tax money. Most don't, so you can ignore this. Maximum here is $36000.
4. Once you've exhausted your tax-advantaged options, taxable brokerage account at one of the high quality low cost brokerages (Vanguard, Fidelity, Schwab).
In all the above accounts, hold a mix of broad, low cost mutual funds or ETFs. The exact mix is easily debatable (see
150 Portfolios Better Than Yours | The White Coat Investor - Investing And Personal Finance for Doctors for 150 examples). Some funds are better in some forms of the accounts (total bond is very tax inefficient in a brokerage account, but municipal bond funds do fine) but in the long run, just saving enough money and putting it in anything approaching a sane distribution will be just fine. I'd say for the general person shoot for saving 20% of your pretax income, but more or less may be needed depending on your specific goals.