Is there any flying school out there that use "jets" as part of their training program? I mean like a Cessna Citation or Phenom.
It's rare (and EXPENSIVE) but there are a few flight schools which offer primary flight training in the Embraer Phenom 100 and Citation Mistang aircraft. Angel City Flyers in Long Beach, CA (KLGB), is one such flight school. Bring your banker; it's going to cost $1000+ per flight hour plus fuel and instructor fees!
Another place is the National Test Pilot's School in Muroc, CA. Their 11 month Masters of Science in Test Piloting course will make you an accredited test pilot and log flight time in over 30 aircraft, including jet fighters like the Saab 35 Drakken. Tuition is $960,000!!!!!!
FlightOptions at the Sunshine Coast in Australia have a Level D Citation Mustang simulator, and a real Mustang as well. The facility used to be used by Singapore Airlines for their pilot training. Great location!
The Swedish Air Force uses the Sk 60 jet (Saab 105) as its only trainer, including initial training.
For initial training, I don't think there is any flight school or Air Force in the world that will let you begin in a jet.
At the later stages however there are some programs that incorporate jet time. I believe Lufthansa's ab initio course uses a Citation as the final stage before joining the airline. But generally speaking most pilots in training programs don't fly jets until joining the airline.