Small piston airplanes are "cheap" (as airplanes go) to operate and jets are an order of magnitude more expensive. For example, you can find small piston trainers on the order of \$100/hour to operate whereas a jet is often well north of \$1000/hr to operate. If your wallet can accept this, this is just one barrier to starting training in a jet.
Small piston training airplanes are everywhere. Go down to your nearest airport and you'll probably find a few. Jet airplanes for training for the most part just do not exist. The vast majority of jet training happens in simulators, and these simulator courses are really only designed to provide type ratings and possibly ATP ratings, not primary training. This means you'll probably have to buy or lease a jet. This will be very expensive. This is your biggest barrier to starting training in a jet.
If you own or rent a jet, you want to insure it. I sincerely doubt you'll find anyone willing to underwrite hull loss insurance on a jet used for primary training of an unrated pilot. Perhaps you could find coverage for a premium equal to the full replacement cost of the jet, but that really isn't practical. This is a barrier to training, but if you can afford to do this you might not care that you can't find insurance.
Speaking of type ratings, it is very likely you are going to need one of those to operate the jet, this will complicate your training and checkride because type ratings are tested to ATP standards, which means you will be required to fly to a higher standard with tighter tolerances than you would otherwise be requires at this stage of your training. You'll also be crippled without an instrument rating because you can't fly the jet above 18,000 feet (USA) without one.
Speaking of checkrides, you'll need to find an examiner that is typed in the jet you are using and who can conduct the checkride. Easier said than done. A typical jet isn't really suited to the private checkride either. The required cross country navigating with a map and the required diversion would be a lot of work in a very fast and complex airplane for a brand new pilot.
Before you can go on a checkride, you need training. You need a CFI who is typed in the jet and that CFI is going to be more expensive than the CFI you would be using in the piston airplane. Probably much more expensive. You will probably also need to spend much more time with the instructor than you would in the piston airplane because the jet moves much faster and you will be flying to ATP standards instead of private pilot standards. More practice == more money.
If you want to be in a jet ASAP and you can afford to own one and insure it, your best bet is to just do the private in a piston airplane then do an instrument rating in a piston airplane, do your multi-engine rating in a piston airplane, and then get a type rating in the jet to transition into flying the jet. You'll probably also need to hire a typed pilot to fly with you (for insurance costs) until you build a significant amount of time in the jet. If you didn't buy a jet capable of single pilot ops, then you'll need to hire a pilot because the plane requires two pilots anyway, but the entirety of my post assumes you picked a single pilot capable jet.