I took a look at a verb conjugation dictionary and they appear consistently identical, but I was wondering if I could just remember it as a rule that the verb conjugation forms for 'il' and 'elle' are always the same and the verb conjugation for 'ils' and 'elles' are always the same as well.
Yes, other than in the case of compound tenses (temps composés) that use être. Notice that in the TLFi morphology entries for verbs, the forms for a particular tense are listed only for variations in number (singular or plural?) or person (first, second, or third?). The exception to that is in its conjugations for the past participle, the inflections for which depend on gender as it may function as an adjective or be used with être. For example, réveiller may be reflexive, or it may be applied to another object:
Se réveiller: Il s'est réveillé à 6h. OR Elle s'est réveillée à 6h.
Réveiller: Il l'a réveillé. (He woke him up.) OR Il l'a réveillée. (He woke her up.)
It depends on whether you're distinguishing conjugation from the broader phenomenon of inflection.
Strictly speaking, the finite components, the conjugated verbs, are always the same:
il/elle voit ; ils/elles voient ; etc.
But some compound tenses also contain past participles, which can be inflected for gender and plurality:
il s'est trouvé, elle s'est trouvée
ils se sont changés, elles se sont changées
Note that the finite verb (est, sont) does not change.
Nor does the stem ever change in such cases — only the agreement, the e and/or s at the end. Moreover, unlike adjectives, participle agreement is 100% regular (to my knowledge).
So in a narrow sense, il/elle and ils/elles do always share the same conjugation. However, compound tenses may include elements that show agreement. But that's a matter of learning some grammatical rules; you never have to memorize extra base forms for il/elle and ils/elles.
Those rules are, I think, outside the scope of this question, but here's a handy chart for reference:
For simple tenses (like present), you are right.
Il/Elle part. Ils/Elles partent. Ils/Elles viendront.
For compound tenses it may be more tricky. For verbs conjugated with avoir as auxiliary:
Il(s)/elle(s) a(ont) mangé (same forms between masculine and feminine).
But for verbs conjugated with être as auxiliary there is a difference.
Il(s) est(sont) parti(s) vs Elle(s) est(sont) partie(s).
So, one has to learn which verbes are conjugated with être in compound tenses. in brief, the mnemonic rule is MRS DR VANDERTRAMP. (see here: https://www.fluentu.com/blog/french/dr-mrs-vandertramp-verbs/)
NB The "avoir" auxiliary is more tricky : "Elles ont mangé des pommes" but "Les pommes qu'elles ont mangées". Rule : Il n'y a accord du participe passé avec l'auxiliaire "avoir", avec le complément d'objet direct (direct object: COD) que si ce dernier le précède". Here the COD is "que", which is a référence to "les pommes". Warning: with the COD, these are the apples that trigger the female form : "La POMME qu'IL a mangéE"