It depends on your use case. 12400F is good enough for gaming. 13400F is barely 3-2% faster than 12400F on a good day; however, more e-core gives an edge in productivity software. 13500 is a 10-8% faster gaming and productivity beast for the price, but you need a beefy dual tower air cooler to keep it cool.
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It depends a lot on your budget. In my opinion the 13400f is worth it over the 12400f. But i wouldn't pick the 13500 if you use it only for gaming, if you want something more powerfull you might as well pick a 13600k.
I wouldn't cheap too much on the mobo, something like a DDR4 B760 board would be a good pair with enough head room if you want to put a better 13 or 14 gen cpu in the futur.
One thing to also consider is that the 12400f is likely good enough for now. In a couple of years if it's struggling a bit, you'd have a decent number of options for used LGA1700 CPU's that would just drop right in.
In games where you will be GPU bound (which will be most games if you don't have like a 4080 or 7900XTX) these two CPUs will offer almost identical performance.
In games where you a CPU bound, the performance difference will be like 5%
The 13400 is not a very big upgrade over the 12400 and not worth it for the extra premium.
Where as the 13500 has increased cache and is like 15% faster than the 12400 and the 13600K would be about 25% faster.
So my pick would be the 12400F as the intel 6 core budget king, or 13500/13600K if you had the extra money.