The general rule that I learned is: if it comes before the root, it's a prefix; if it comes after the root, it's a suffix; if it comes inside the root itself, it's an infix; if it comes both before and after, it's a circumfix.
Thus, your am-er-emo would be root-suffix-suffix, even though one comes closer to the verb than the other.
On the other hand, this isn't universally consistent. In Swahili, for example, tenses are indicated by a verb prefix, and any prefixes that come between the tense and the root are instead called "infixes". (A-na-ye-penda 3SG-PRES-REL-love "the one who loves" is analyzed as prefix-prefix-INFIX-root.)
In general, more and more linguists seem to be turning to the generic "affix" for all four cases.
EDIT: As requested, a bit of explanation for infixes. The best example I know of is from Proto-Indo-European and several of its descendants. In PIE, an -n- could be infixed before the last consonant of a verb stem to indicate the present tense. Latin, for example, has the verb root vic- "to conquer", as in veni vidi vici. In the present tense, however, it appears as vinc-, as in invincible. Similarly, the root scid- "cut" appears as scind- in the present. (There's actually one and only one instance of this nasal infix remaining in native English verbs: present-tense stand versus past-tense stood.)
EDIT: I forgot interfixes! Interfixes are entirely separate, they're not morphemes at all (in the sense of "meaning-bearing units"). Interfixes are sequences of phonemes that are inserted between two morphemes without having any meaning themselves. English doesn't really have these at all, but in German you see this with the /s/ added between elements of compound words.