I have a Roth IRA and a Roth 403b through Fidelity. For the IRA, I know I can put money at any time from my bank account into it. However, I don't know how to actually do it for a Roth 403b. Is the only way some kind of monthly transfer when I get my paycheck?
No, you cannot put money into a 403(b) plan (or a 401(k) plan) from your bank account; the money has to come from payroll deductions made by your employer (who sponsors the plan) from your paycheck, and sent directly to the plan administrator for depositing into your 403(b)/401(k) account or investing into whatever mutual funds that you choose from those that the the plan offers. That is, you do make a monthly transfer into your 403(b) (assuming you get paid monthly) but not out of your take-home pay or other assets; the contribution is deducted from your wages and sent directly from your employer to the plan administrator. Since this is a Roth 403(b), the money contributed to the 403(b) is taxable income to you, and your employer will be withholding income tax on this money as well and sending it to the IRS, which will further reduce your take-home pay in comparison to what you would get if you were using a Traditional 403(b) account.