I want to clarify first:
Your refund is what you get back if you pay into the system more than was withheld. You can easily get a larger refund by having more withholding (less take home pay each check), but I don't think that's what you want. You want to reduce your overall tax liability -- what you want the government to keep at the end of the year, net after any refund.
Roth IRA contributions are not tax deductible and don't reduce your taxable income. You pay tax on the money that you put in a Roth, so that your withdrawals are tax-free later. If you contribute to a traditional pre-tax IRA, that is tax deductible now, but you pay tax when you withdraw. Whether you use a Roth IRA, a traditional pre-tax IRA, or both, is up to you, and the decision is mainly about when you want to pay the tax, now or later.
403b contributions could be pre-tax, after-tax, or a combination, depending on your choice and the options offered by your employer. Like IRAs, contributions to an after-tax (designated Roth) 403b account do not reduce your taxable income now, but your later withdrawals are tax-free. Contributions to a pre-tax 403b account do reduce your taxable income (your W-2 income is reduced by your employer) and will reduce your tax. However, that also means you are locking up money that you can't spend until you retire. For every $1000 you contribute now, you will reduce your tax liability by $200-$300, depending on your income and your state income tax rate.
Child care is eligible for a credit if you pay for care so that you can work or go to school full time. Babysitting for date night is not eligible. If you are paying for child care so you can work, you should obtain the tax ID number of the provider or providers using form W-10, so you can claim the credit later. However, the maximum expenses you can claim are $6000, so if you already pay that much to your regular day care provider, there would be no extra benefit to getting the same information from your occasional babysitter, even if the babysitting was qualifying care for the credit.
https://www.irs.gov/forms-pubs/about-form-w-10
