What are some points showing Juliet's maturity?

Arguably, Juliet demonstrates rational and mature thinking, but this maturity is, for the most part, demonstrated before she gives her heart to Romeo. 

In Act I, Scene 3 of Romeo and Juliet, Lady Capulet asks her daughter if she can consider Count Paris as a husband; she also informs Juliet that Paris will be at their feast. Juliet replies that she will take a look at Paris and try to like him if what she sees is worth liking. But, she adds that she will not let herself fall for Paris any more than her mother's permission allows. In other words, Juliet has a sense of moderation at this point:

I’ll look to like if looking liking move.

But no more deep will I endart mine eye

Than your consent gives strength to make it fly (1.3.99-101)

Certainly, this sense of moderation demonstrates a degree of maturity.

Further, in the famous balcony scene of Act II, Scene 2, when the passionate and impetuous Romeo swears his love and desire for Juliet, it is she who urges caution:

Oh, swear not by the moon, th' inconstant moon,
That monthly changes in her circled orb,
Lest that thy love prove likewise variable. (2.2.116-118)

Then, when Romeo asks what it is that he should swear by, she tells him not to swear at all. Instead, she urges caution, saying that her agreement to a betrothal is

...too rash, too unadvised, too sudden,
Too like the lightning, which doth cease to be
Ere one can say "It lightens." (2.2.118-120)

Juliet maturely adds that their love is like a bud that will prove to be a beautiful flower if it is allowed to ripen, so they must be patient.

Despite her rather mature and cautionary behavior in the beginning of their relationship, Juliet is later influenced by Romeo's passion, and her love transforms into a "violent delight." That is, the love of Romeo and Juliet becomes the dangerous love about which Friar Lawrence prophetically warns Romeo when he tells the passionate lover,

These violent delights have violent ends.
And in their triumph die, like fire and powder
Which as they kiss consume. (2.6.9-11)

So, while Juliet is cautionary and sensible initially in her approach to love, she later grows more impetuous in both her feelings and actions after secretly marrying Romeo.