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Hello there,

I don't understand what the meanings of some phrases are in the following sentences.

If anyone would know, would you explain them a little bit for me?

Thanks a lot in advance~!

pb.

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I took Melle up on a second free visit to correct my hair color, which lightened significantly, and asked if she could sharpen the wedge.
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When you accept someone's offer, you take them up on their offer.

Can I give you a ride to the airport tomorrow?

Thanks, I think I may take you up on that, if my husband has to work.

Apparently Melle is the hair dresser and the color wasn't right? So Melle offered another appointment at no charge to fix it, and the person writing this accepted that offer.
Comments  
This sounds like a woman or girl talking about a haircut or hair style.

The "wedge" is a new style, where the bangs, front sides are longer than the back. The back is cut up short and that look, plus the longer sides looks like a like a "wedge".

Some like the wedge to be very short or "sharp"; others like the wedge to be just slightly slanted.

Tom Cruise's wife has worn her hair in a wedge style. It is popular in Britain, also.
 BarbaraPA's reply was promoted to an answer.