Translation:If you go to the movies, take your sister.
For the same reason llevarse is used for "to wear". When you carry something on your person, you use the reflexive pronoun. Metaphorically, you're being told to carry your sister with you -- to keep her safe. That kind of protective big sibling / little sibling relationship is implied with just the single added syllable. It's actually kind of a neat feature of the language.
If you were just going to drive your sister to the movies and drop her off, it could just be lleva.
llevar can be used to take something somewhere, but in that case you need to specific the location. llevarse can just be used to mean to take something with out, without mentioning the destination. I think that is the reason for the pronominal version here. It doesn't matter grammatically, I don't think, that the movies is mentioned in the prior phrase.,
The indicative is used for both the “if” and “result” clauses if it is a “real” condition. In an “unreal” or “contrary-to-fact” condition, a past subjunctive is used in the “if” clause, and normally the conditional is used in the result clause.
That's according to this
Note: Scroll down to "VI. 'If' clauses"
Here is a great discussion specifically related to your question ("Si vaya…"). Despite the advice (which is, of course, a sound one) seen everywhere that says "never use the present subjunctive after si!", someone quoted Lazarus (who's a god [for me] when it comes to Spanish grammar in forums) in that discussion:
"There are rare cases where present subjunctive can be used afer "si", but leave them until you are fluent. Many natives have never seen them before."
The alternative would be to use past-subjunctive: "Si tú fueras al cine..." That would mean something like, "If you were to go to the cinema..." or, "If you went to the cinema..." -- but not in a past-tense sense, in the sense of, "If you went to the cinema right now, I would feel sad, because I thought we would go together tomorrow." Si tú fueras al cine ahora, me sentiría triste, porque pensé que iríamos juntos mañana.
The present subjunctive is never used after si.
If something is really uncertain you can use the past subjunctive
yes I know, using the past subjunctive for a future hypothetical
event might not make much sense but actually the same thing
is done in proper English.
If I were to win the lottery, I would quit my job.
This is past subjunctive in English and then same can be done in Spanish
Si yo fuera a ganar la lotería, renunciaría mi trabajo.
However, the subjunctive should only be used, both in Spanish and in
English if the possible of occurrence is seen a low / doubtful.
That is why it isn't used in the present sentence.
The probability that he goes to the movies is seen as quite possible.
When verbs are in their original (infinitive) form, they end in "r". We all know that any word ending in a consonant other than "n" or "s" is naturally stressed on the last syllable -- "lleVAR" ("cociNAR"). However, when a verb is conjugated, its ending becomes a vowel (or an "s" for "tú" and "vosotros"), so now the natural stress must be on the second-to-last (penultimate) syllable according to the rules -- so "lleVAR" becomes "LLEva" ("cociNAR" becomes "coCIna"). But we need to add one more syllable now -- "te"; however, we also need to keep that stress on the "LLE" part of the (conjugated) verb itself ("te" is just a pronoun attached to a verb, but it's not part of the verb). So to do that, we need to write the accent; otherwise, it would naturally move to the new penultimate syllable "va" and the word wouldn't be pronounced correctly.
Emory, 'llévame al hospital' - as you say the 'me' is the direct object. In 'llévate', the 'te' is an indirect object which would translate to 'with you', but DL has not translated it. My step-mother used to say 'wash me these dishes' which meant 'wash these dishes for me'. She spoke a regional type of English. The reason you cannot say 'llévala a tu hermana' is because in Spanish, you need either the direct object pronoun or the direct object noun, but not both. You could say 'llévala al hospital', take her to the hospital, or 'lléva a tu hermana...' take your sister to the hospital, but you would then be using the verb llevar instead of llevarse. The latter has that extra nuance that the sister is going with you. An indirect object pronoun and a direct object noun, which go very well together.