If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
If you claim the child, you may use Head of Household (HoH) filing status. If you do not claim the child, you must use Single filing status. Your girlfriend must use Single filing status whether she claims the child or not.
You must claim the child, as a dependent, to claim the dependent care (day care) credit.
