If I have "mi legas libronj" (I read books), does -n or -j come first? Does it matter? Is there a standard? What about other suffixes? (I've only learned objects and plurals so far but assume there are others)
These are actually not suffixes, but endings. The plural ending -j is always written before the accustaive -n. Other endings are -o (for nouns) -a (for adjectives), -e (for adverbs) and the verb-endings. I don't think you'll be confused by them.
Later when you learn the suffixes, (-et, -eg, -in, -id are some of the first you learn) you'll see that they always come before the endings. For example: dom-et-o-j-n. Occasionally, you have to be careful with the order, but this is not something one runs into too often.
Yes, it very much matters. The -j must come first. Therefore, you cannot say "libronj" rather "librojn". There may be some rationale besides pronunciation (the former has one extra syllable- "li-brau-ni" VS. "li-brauyn"), but I'm not aware of it.
As far as affixes go, the order can very in relation to meaning trying to be expressed.
malsano- unhealthy, sick
malsanulo- unhealthy, sick person
malsanulejo- Place for unhealthy people, hospital.