Circuit analysis with 2 diodes : Constant Voltage model
But how can I know which of 2 diodes will turn on first as V_in
increases from -inf?

Well there are tools that can help such as a simulator but I guess you might not be that keen on simulation so, ask yourself, what maximum positive voltage can be at the input that doesn't cause current to flow through D1. No current in D1 means you can analyse stuff without it affecting the math.
I say a positive voltage because clearly D1 is reverse biased by \$V_B\$ so, to make it conduct, D2 and R2 must be acting as a partial clamp on the output so that there is a volt drop across R1.... But there's the answer, D2 has to be conducting before D1.
Alternatively you could look at the problem without the presence of D1 and, in your minds-eye, raise the input voltage and tabulate what the voltage is across the nodes of the missing D1 diode. At some point it will become forward biased and, at that point that is the voltage needed at the input to get to the onset of D1 conduction.
But, in short, D2 will be conducting before D1.