When it is really cold I cannot get the car into 1st gear or 2nd gear. I had to
let the car warm up for about five mimutes before I could drive. Then I was
smelling a burning odor. Now, I notice the colder it is the harder it is to get
into gear.

Is my clutch dying?
Is it expensive to fix?
Can I fix it myself?

 

check to see what type of fluid is in the transmission. If the oil is thick, it may be hard to shift. If the oil is the proper lite weight, it should not have that hard shift. my 02 civic in big bear ca. skiing, them was 5 degree all night, next morning, warmed up enough to clear my windows and drove up to the slops to ski, no shift problem. all worked well. clutch job about $350.00 for my car. yes you can fix it yourself if you are mechanical minded and view a few of the U tubes, also have a manual to help you.