Learn Arabic (Modern Standard): أبي يحبني أيضا.

The base verb is يحب (he loves). When we attach a pronoun for "me" to a verb, Arabic uses ـني (-ni) instead of just ـي (-i). The extra "n" sound stops the verb's final vowel from being changed or swallowed by the heavy "ee" sound of the pronoun.

Does this mean "my" on nouns is different from "me" on verbs?

Exactly! For nouns, you just attach ـي (-i) to mean "my", like the word أب (father) becoming أبي (my father). But verbs always take ـني (-ni) to mean "me", like in يحبني (he loves me).

Is it normal to put the subject أبي before the verb يحبني? I thought Arabic verbs usually come first.

You are right that Arabic often starts sentences with the verb. However, starting with the noun is also perfectly correct and very common. Putting أبي first simply puts a little more focus on the subject.

What are the two little diagonal lines above the end of أيضا?

That is a pronunciation mark called "tanwin". Specifically, it is a double fatha. It adds an "-an" sound to the end of the word, which is why the word is pronounced "aydan" (also/too). You will see this specific "-an" ending a lot on Arabic adverbs.