First off: If you intend to record chamber music, don't go with a surface-contact microphone such as a piezzo microphone. These microphones are attached very very close to the instrument, and you will not get the sound you want to have. (Remember that that is not where you put your ears when you listen to the result.)
I have no idea what you mean by a 'classical' microphone, please enhance us with a brand name and type number.
A solo cello is best recorded with a large diaphragm condenser microphone such as the Neumann U89. I understand though that you are on a budget, so you might consider a more affordable alternative instead. In the past, I got pretty good results using the Studio Projects B1 microphone, which sells for around $100. It needs a general microphone pre-amp that is capable of feeding 48v phantom power.
If you are recording a chamber music session (e.g. a piano trio or a string quartet), you might however get better results by not trying to record all instruments individually, but getting the entire sound using a stereo pair of microphones. In that case, go with a pair of small diaphragm condensers. For affordable solutions, I can recommend a matched pair of Oktava MK-012 (Made in Russia, beware of Chinese fake) or a matched pair of Rode NT5s, both ship for around $300. These too need a preamp that is capable of giving 48v phantom power.