If the county collects 2024 property taxes in arrears, 1st half due 1/31/25 and 2nd half due 6/20/25, what amount do I enter for property taxes paid in 2024?

Next year you will enter the 6/20/25 payment plus any other payment for 205 you make in 2025.  

But you might not even need to enter it unless you know you can itemize deductions.  And t tax paid is limited.  For tax years 2018 through 2025, the SALT deduction (which includes sales tax) is capped at $10,000. SALT is State And Local Tax. Which includes property tax, any state tax paid like for last year’s return and includes any state withholding from your W2s and any 1099s you have. And any taxes in W2 box 14 and 19 like SDI or VDI. You can only deduct up to 10,000 (5,000 MFS) for SALT State and Local Taxes.

And other deductions like Medical is limited.   You can only deduct the amount of unreimbursed Medical Expenses you actually paid over 7.5% of your AGI. 

Then all your deductions have to be more than the Standard Deduction to be used. 

For 2024 the standard deduction amounts are:

Single 14,600 + 1,950 for 65 and over or blind (16,550)

HOH 21,900 + 1,950 for 65 and over or blind (23,850)

Joint 29,200 + 1,550 for each 65 and over or blind (30,750/32,300)

Married filing Separate 14,600 + 1,550 for 65 and over or blind (16,150)