I like to start with Blues for Alice. It's a blues for one. It is also a bebop tune and learning lines like these is a good way to start to break down and change some bowing habits you might have, as well as challenging your idea of what an appropriate sound is. If you are a classically trained cellist like I am you've spent years with a set of expectations that need to be jostled and challenged.
Phrasing & Time
Learn "Blues for Alice" Try to use the bowings supplied here. If you achieve a better, more convincing sound using a different bowing, please do. If you're not sure then stick to the bowings I have indicated. It's not a bad idea to get a hold of a recording of this tune before or while you are learning the notes and the bowings. Knowing how it should sound will help you understand my bowing choices. "Swing" is an elusive concept. For now think of a dotted eigth-sixteenth note pattern or quarter-eighth in a 3/8 bar as a starting point. One teacher of mine put the words "oooh-bah" to those rhythms as a way of hearing a swung pair of eighths.
Start slowly and work up to tempo. Use NO vibrato. When approaching improvised music your "sound" is a very important consideration. If you are classically trained then you're pretty sure to be using vibrato quite freely--some teachers even demand its use on every note you play! The rich, beautiful classical sound that we are trained to have should be only one of the colors in your arsenal (for any kind of music.) In most improvised music this kind of sound will stick out and make you sound stiff. So eliminate vibrato for now. Use bow speed and contact point (near or far from the bridge) to vary the color of your sound.
Use a metronome. Start slowly and set it to click on the "2" and the "4" of the 4/4 measure. This is a good way the feel of a "jazz" walking 4/4 feel which, for the sake of a simplified learning environment, I will say doesn't stress the "1" and the "3" (nothing is a hard and fast rule. As you get better and are more at ease you can emphasize different beats to create variety. For now stick to feeling the "2" and the "4" as the stronger beats.
LISTEN AND PLAY: Get a recording of Blues for Alice (the tune is on the recently re-released "Charlie Parker Swedish Schnapps +" cd on Verve) and play along with it. Listen to the phrasing. Listen for ghosted notes (notes that are "swallowed" or not fully articulated.) Listen for how the eighth notes are phrased...are they swung eighths? If so, are all the notes swung? Or are some straight eighths? (My feeling is that an incessant swinging eighth note feel is artificial and stiff. Don't become locked into a "triplet-y" swing feel, this will sound lame. Your phrasing should be fluid, changing , varied. [As is often the case the "Real Book" has a version of the tune with some notes that are played differently by the band on the recording. Test your ears by finding the "errors" in the chart.]
Play the head even while listening to the players on the recording soloing. Always keep the form of the tune.