Learn Italian: Domani la comincio a lucidare. - Tomorrow I am starting to polish it.

With aspectual verbs like cominciare (to start), you have a structural choice called clitic climbing. You can place the pronoun before the conjugated verb (la comincio a lucidare) or drop the final -e of the infinitive and attach it there (comincio a lucidarla). Both are completely correct and mean the exact same thing.

Why do we need the preposition a in this sentence?

The verb cominciare requires the preposition a whenever it is followed by another verb in the infinitive. To say 'start doing something' in Italian, you must use cominciare a.

Since the sentence starts with Domani, why is the present tense comincio used instead of the future tense?

It is very common in Italian to use the present tense for near-future events when the timeframe is explicitly stated, like with Domani (tomorrow). Saying la comincerò a lucidare is also correct, but the present tense feels more immediate and natural in spoken language.

What does the pronoun la tell us about the object being polished?

It tells us the object is grammatically feminine singular, like la macchina (the car) or la scarpa (the shoe). If the object were masculine, like il tavolo (the table), the sentence would use lo instead: Domani lo comincio a lucidare.