The core +damage mod is the most important, it actually modifies the weapon's base damage, where as Fire, Ice, Elec. and AP all add damage based on the weapon's damage. So a 100 dmg gun with a +20% Weapon Damage mod...means the Fire Mod that adds 20% Fire damage would add more damage to each shot from a weapon with a Weapon Damage mod...versus one without.
Also research your Warframe's basic stats. You might find, for example, that the low armor on many Warframes make the Armor mod a poor choice versus other mods (Sprint Speed, Power Damage, Power Range, Energy Cost Reduction, ect....) Or for example, you might find your WF has a sick base health but crappy base shield....making you question if it's even worth it to use the Shield Capacity mod. Health is pretty much non-negotiable though, unless you've got insane shield values and never let your shield drop.
My advice for grinding mods is the lower level defense missions. Go to Wave 15 on those and you can walk out with 10+ mods easily. Granted, most of those will not be for your Warframe or mods you're interested in using, but they're all grist for Fusion so you can level the mods you are using. As a plus, Defense Missions tend to offer either Void Keys or some of the good mods (read as: damage and elemental mods) as the wave rewards. There is also a loose correlation between mission level and the kinds of mods you get....but it's very hard to pin down.