Good question. First let's recognize that thiols (-SH) are more acidic functional groups than the corresponding alcohols (-OH) (some rationale here). So if we were talking about pKr we would see around 8 for cysteine's thiol side chain and around 15 for serine's alcohol side chain.
Now, to address your question about the carboxyl acidity, pK1, of these compounds. Serine's OH side chain has the more electronegative atom than cysteine's SH group, so we would indeed expect a stronger electron-withdrawing inductive effect in serine, possibly increasing the carboxyl acidity. However, the O atom is two carbons away from the carboxyl group, so this effect may be minimal to negligible... Not to mention of course the empirical data you provided implies that cysteine has the more acidic hydroxyl, not serine.
So I'm just speculating without looking at the actual experiments used to establish the pK1s, but it might be possible that when attempting to measure the pK1 of cysteine's carboxyl, the acidity of its thiol may have slightly skewed the measurement to make the carboxyl appear more acidic. I'm not sure if this was controlled for in the pK1 determination, but either chemically or computationally removing the contribution of the thiol to cysteine's acidity might result in a slight discrepancy compared to what it "really" is.
Additionally, one might imagine that the acidic thiol of could intramolecularly protonate cysteine's amino group, creating a proton-labile positive charge (-NH3^+) quite close to the carboxyl, and this might serve to stabilize the negative charge of the carboxylate conjugate base (-COO^-) thus increasing the carboxyl group's acidity. This effect would be more prominent in cysteine (due to the acidity of the thiol) than in serine (whose hydroxyl would be less likely to protonate the amine to stabilize the carboxylate). So this is a possible (but perhaps oversimplified) mechanistic rationale of why cysteine's carboxyl is more acidic than serine's.
As for the amino group, being more often protonated by the side chain in cysteine, it may also appear to be more acidic in serine.
This is just conjecture, for a complete understanding I would recommend looking at some experiments in the literature to see how they handle determining pK1 and pK2 when an acidic (or basic) side chain is present. It's possible that common practice is to eliminate the side chain's effect by computational or chemical methods, so without examining the primary sources one cannot be certain exactly what the reported pKa values actually represent.
