Can I use the private pilot written exam for sport pilot? I'm planning to first get my sport pilot license and then continue on to private pilot, to save on written exam costs is it possible to just take the private pilot written exam for the sport pilot license or will I have to take both?
No.  You will have to take both written and practical exams for sport and private pilot, if you go that route.  I’d recommend that, if you do ultimately want to obtain a PPL, that you not pursue an SPL as part of your training and solely concentrate on obtaining your private only.  You can exercise all the privelages of an SPL with a PPL, and it cuts out the excess costs and time with the SPL testing and checkride prep.