The steps in the learning process are always the same, regardless of genre - of course, depending on the difficulty of the piece, the weighing of the categories can change rather drastically!
Starting a new piece in so fun, exhilarating, and pretty soon frustrating. I often do a first, quick pass at fingerings and bowings, just to have something to work from, and assume that these will change as the piece matures. The more I learn, the more I understand that studying the piece without my instrument is essential, and the sooner I do it, the better: I need to know about form, harmony, details of performance instructions...but especially harmony! Starting to pay more attention the actual chord analysis on Bach has essentially revolutionized how I practice, and memorize it now -- every time I find something new and my mind is blown all over again.
My favorite part of the learning process...Drilling. No thinking, just doing and listening, purely physical. Finding the most efficient and elegant way of playing a passage, then solidifying it and making it easy. I love the use of gadgets - metronome, egg timer, tuner, recording devices.
The most difficult part: artistic choices. The more choices there are, the harder it is for me. Luckily, I have had some very excellent teachers and I know I lot about the history of the different cello styles, so a lot of times I already know the route I'm meant to take. Again, new music is in a different category - I usually hope that by the artistic choices time, I have learned to speak the language of the composer, and I know the dialect that I want to speak.
When all is said and done, it's important to be able to play a piece from top to bottom. This is really where the transition to performance starts: how do we string together all the elements into a coherent whole? By practicing sequencing of course. I like to start very slow, so that my mind is always ahead and able to process all the complexities of the piece. Here I also think very actively about relaxing, going to that happy place where my body is so loose that it can react with lightning speed to any commands my brain gives it. Usually, intonation gets better. Funny how that happens.
Finally, transition to actually performing the piece....It would be foolishness to think that one could perform a piece to the top potential with only having learned the piece, not heaving learned to perform it. Top to bottom, no stopping. It is very important to practice the performance situation as well, to simulate the physical response. If I'm not intimidating enough to make my students nervous for a practice performance, I make them run stairs (or around the building) to get their heart rate up. I also often record practice performances, to up the ante so to speak. This transition is definitely not a linear one - we have to learn from all the mistakes, go back to the learning process, repeat, rinse.