Hi There!
If you are looking to get a new mouthpiece it is a good idea to try and identify why - what is missing in the sound that you currently produce, or what sort of sound you are wanting to produce. Some folks find the Yamaha 4C a little too neutral sounding so you may be looking for a more distinctive sound, for example. I would say that some mouthpieces are more suited to classical - narrower tip
and with a "purer" sound, which sound good with slightly harder reeds - such as Vandoren Blue Box Traditionals. Some are more suited to rock, contemporary and similar music, with a wider tip and softer reeds such as La Voz, Vandoren Java. The Third are mouthpieces more suited to Jazz - again wider tip - which are usually based on Meyer type mouthpiece design, and use Jazz Cut reeds, such as Vandoren ZZ, Marca Jazz etc. A bit of research would be helpful on your part before you visit.
With something like Vandoren, Classical would be the "Optimum" mouthpiece, Jazz would be the "V16" mouthpiece, and Rock/Pop would be the "Java" or "Jumbo Java". If you are visiting a shop to try a few mouthpieces I would suggest having a try of some different styles of mouthpiece before trying to make a choice.
As far as exercises are concerned, just do some basic stuff - long notes, scales, short phrases and similar to see what you think of the sound produced and whether it feels comfortable to play, etc. I would think about one or two pieces that you play regularly to see whether the sound is in the right area. I primarily like jazz music from the 60's onwards and my main pieces are jazz suitable, Meyer type mouthpieces with Jazz Cut reeds. At the same time I like some Rock/Pop stuff and have a slightly different mouthpiece/reed set up just for that purpose.
In my experience many people have a good sense fairly early on of what mouthpiece they like the sound of, so exercises are not that important, but having a piece that you know well and characterises what you like about playing sax will help you choose a mouthpiece which helps you produce the sound that is nearest to your heart. Next is to get the right right tip opening size and right reed strength/type.
Good luck!
Tom