Resistors in series pass the same current and share the voltage, so their resistances add. Resistors in parallel see the same voltage and share the current, so their conductances add — which means you sum the reciprocals.
Two useful shortcuts. For exactly two resistors in parallel, R = (R1 × R2) / (R1 + R2), the “product over sum” form. For N identical resistors in parallel, R = R_one / N.
Sanity checks
A series total is always larger than the largest resistor in the chain. A parallel equivalent is always smaller than the smallest one. If your answer breaks either rule you have inverted something. A parallel combination of 1 kΩ and 1 MΩ is about 999 Ω — the big resistor barely participates, which is why a high-value pull-up beside a low-value path is effectively invisible.
Getting values you cannot buy
Standard E24 parts jump in steps of roughly 10%, so exact values often do not exist. Two 10 kΩ resistors in parallel give 5 kΩ; in series they give 20 kΩ. Adding a large resistor in parallel with a smaller one trims it down slightly, which is a common way to hit a precise divider ratio without buying 0.1% parts.
Power still has to be checked
Combining resistors does not combine their power ratings usefully. In a series chain the largest resistor dissipates the most, because it drops the most voltage at a shared current. In parallel the smallest one dissipates the most. Check the worst-case part, not the average.