I know exactly what you'd be doing musically.

You would almost certainly play the virginal, a type of harpsichord where the string is plucked towards the center, giving it a very different tone--slightly more harplike but still with a harpsichord's twang. Virginals were considered appropriate for young ladies because there was nothing suggestive about playing them, and they were played standing, and therefore there could be no accidental showing of the leg.

Its popularity with young women can be seen in the titles of many of the collections of virginal music that we still know of. A majority of them were dedicated to or originally written out for a specific noblewoman: Elizabeth Roger's Virginal Book, My Ladye Nevells Booke, Parthenia [a classical reference to young maidens, also the first printed collection of virginal music in England], Priscilla Bunbury's Virginal Book, Anne Cromwell's Virginal Book

You would also probably play the treble viol in consort with friends or family members as many wealthy families owned a chest of viols (a term for six viols and the bit of furniture to house them in, usually in a set of 6 by the same maker and matched to each other so as to stay in tune and blend better) --never the bass viol, as that is held between the legs with the knees flung wide and thus not terribly appropriate. Recorder is another option.

It's a pity that it isn't 1659 because then you would be able to read Christopher Simpson's fantastic text, "The Division Violist", and it's followup in 1665, "The Division Viol-or The Art of Playing ex tempore to a Ground", which is an amazing instructional text on viol playing and on proper improvisation techniques to be used over a ground bass--a term for a repeated bass line with or without figuring (a system of indicating the harmonies intended by the composer--consists of small numerals and some symbols written over the actual composed bass line), sometimes with a basic melody included. The term "division" just means "variation".

You would be intimately familiar with the music of John Dowland, who would have died 24 years previously. He was a famous lutenist and songwriter whose music took all of Europe by storm. His most famous song is called "Flow My Tears", also referred to in instrumental form as "Pavana Lachrymae", and was an international superhit, so to speak. Composers throughout history ever since have set their own versions of his song, and Dowland himself did a few different settings and transcriptions for various instruments. Another of his biggest hits was "Come again" which some people think may have suggestive lyrics. ("to die with thee again in sweetest sympathy"--could refer to "le petit mort", or an orgasm. Or not.)

Sources: Dowland, John - "Tbe Firste Booke of Songes", London, Peter Short, 1597 Dowland - "The Second Booke of Songes or Ayres", London, George Eastland, 1600

Secondary sources: Christina Bashford. "Chamber music, §3: 17th and 18th centuries (i) 1600–1740.", Grove Music Online, ed. L. Macy, grovemusic.com

Hawkins, John (1776). A General History Of The Science and Practice Of Music: In Five Volumes, Volume 4, p.339-40. Payne.

So /u/citrusonic has a really great answer with regards to music, but what if you had no talent for music or were not interested in playing it? What other activities would the daughters of English nobility take part in circa 1650? How much and what type of education would they receive?

I've also seen in many period movies and TV shows that young women would be sent to the court of a higher ranked noble (the King, Duke, etc) to be a lady in waiting. At what age did this usually take place? How much training was a new lady in waiting expected to have before she could be accepted by whoever she is serving?