How do I figure out how many hydrogens my compound actually has using a mass and NMR spectrum?
Being an NMR fan myself I would inspect that NMR spectrum:

 

The integrals suggest you have 11 $\ce{^1H}$ or a multiple thereof (the number under each peak is the normalized integral, which is proportional to the number of protons represented by the multiplet). That leaves you with $\pu{87 Da -11 Da}=\pu{76 Da}$ to explain. If you throw in an oxygen you get $\pu{60 Da}$, which is neatly divisible by 12, for a formula $\ce{C5H11OCl}$. Next use the NMR spectrum again to infer the connectivity. The rightmost integrated multiplets (multiplet and triplet) are typical of $\ce{-CH2CH3}$ with $\ce{CH2}$ split by methyl and something else. From the integrals we see clearly there is only one methyl group, and there are no lone H which seems to eliminate the possibility of carbonyl or hydroxyl groups, suggesting an ether. 

In the end I just "cheated" (it's 2019 after all) and used the online NMR simulator predictor:



This nice piece of software lets you tinker quickly with different structures and inspect assignments interactively, helping you develop your intuition.

Next you need to verify the MS and IR spectra, but the NMR spectrum is a giveaway that this is the correct hit.