Dr. Kaku (
http://en.wikipedia.org/wiki/Michio_Kaku ) built a particle accelerator himself in his last year of highschool. He got his parents to help him make solenoids from the copper wire he bought. They didn't know exactly what he was building but knew it was something important. So yes it is possible to build a simple particle accelerator.
To detect "natural" neutrinos (coming from the Sun, not those around nuclear reactors) they ended up building huge tanks full of cleaning fluid (which contains chlorine 37 which reacts with neutrinos to form argon-37) in deep mine shafts. The largest of these capture around 10,000 neutrinos a year, so any argon-37 is difficult to detect.
Building a muon detector may be more feasible. Here are some links explaining:
http://hps.org/publicinformation/ate/q3781.html
this one even has some instructions on how to build one:
http://hps.org/publicinformation/ate/q3781.html