Extreme-UV and X-ray emissions from stellar coronae drive mass loss from exoplanet atmospheres, and UV emission from stellar chromospheres drives photo-chemistry in exoplanet atmospheres. Comparisons of the spectral energy distributions of host stars are, therefore, essential for understanding the evolution and habitability of exoplanets. The large number of stars observed with the MUSCLES, Mega-MUSCLES, and other recent HST observing programs has provided for the first time a large sample (79 stars) of reconstructed Lyman-alpha fluxes that we compare with X-ray fluxes to identify significant patterns in the relative emission from these two atmospheric regions as a function of stellar age and effective temperature. We find that as stars age on the main sequence, a single trend line slope describes the pattern of X-ray vs. Lyman-alpha emission for F, G and K dwarfs, but the different trend lines for M dwarf stars show that the Lyman-alpha fluxes of M stars are significantly smaller than warmer stars with the same X-ray flux. The X-ray and Lyman-alpha luminosities divided by the stellar bolometric luminosities show different patterns depending on stellar age. The L(Lyα)/L(bol) ratios increase smoothly to cooler stars of all ages, but the L(X)/L(bol) ratios show different trends. For older stars, the increase in coronal emission with decreasing T(eff) is much steeper than chromospheric emission. We suggest a fundamental link between atmospheric properties and trend lines relating coronal and chromospheric heating. See paper in ApJ volume 902 (2020).