Why was Romeo interested in Juliet?
This question is unclear as to whether it is asking about Romeo's initial interest in Juliet or his continued interest by the end of the play. I will focus my answer on his initial interest.
I think Romeo is initially interested in Juliet because he finds her really attractive. Some people would argue that Romeo's first glimpse of Juliet is evidence for love at first sight. What the text indicates is that Romeo is immediately attracted to Juliet based solely on her appearance. He says that she is too beautiful for this world and far outshines the beauty of any other woman.
Oh, she doth teach the torches to burn bright!It seems she hangs upon the cheek of nightLike a rich jewel in an Ethiope’s ear,Beauty too rich for use, for earth too dear.So shows a snowy dove trooping with crowsAs yonder lady o'er her fellows shows.
Did my heart love till now? Forswear it, sight!For I ne'er saw true beauty till this night.
Well, in that hit you miss. She'll not be hit
With Cupid's arrow. She hath Dian's wit,
And, in strong proof of chastity well armed,
From love's weak childish bow she lives unharmed.
She will not stay the siege of loving terms,
Nor bide th' encounter of assailing eyes,
Nor ope her lap to saint-seducing gold.
One fairer than my love? The all-seeing sunNe'er saw her match since first the world begun.