I've looked all over but can't find a site that will just tell me so I figured I'd ask here, so verbs will often get written in a group like this: amo, amare, amavi, amatum facio, facere, feci, factus
so if amo is "I love" while facio is "I build" and amare is "to love" while facere is "to build"
then what does amavi/feci and amatum/factus mean?
Amō amāre amāvī amātum or faciō facere fēcī factum are called the "principal parts" of a Latin verb. From them all other forms of the verb can be made if the appropriate transformation rules are applied, so by memorizing those four forms you gain the ability to use the verb in all of its forms.
There are 4 different varieties (called "conjugations," as you noted) of regular Latin verbs, (sometimes a fifth is listed, faciō is an example, which removes a couple of rules), which is why the endings are different between the two sets of verb forms you listed, as they are not in the same conjugation.
Here are explanations of the principal parts from two fairly complete Latin grammars: Gildersleeve and Lodge (§ 120) and Allen and Greenough (§ 172) and a briefer grammar, Bennett (§ 99). (Note that the explanation of the fourth principal part differs between the first grammar and the orther two, although the difference only matters in some verbs.) If you look further along in the grammars you can see examples of the different conjugations.
I'm not too familiar on amatum or factus, but I do know that amavi and feci are used for forming the perfect past tense. This is a completed past action. For example, if you add the third person singular ending, "amavit" means "he/she/it loved" and "fecit" means "he/she/it made".