This issue is with an American Standard single piece toilet. I just changed the 'fill valve' and since then the toilet water erupts shoot upwards, making a mess. The amount of water shooting upwards is very little but enough to reach the toilet seat. The Toilet does not seem to be plugged. I talked to the plumber about this, and he suggested using the 'SNAKE' to clear any blockages.
Water probably isn't "shooting upwards", but rather not draining fast enough based on the inflow of water. Per your plumber, it is probably partially clogged. With a toilet, you want to use a toilet auger instead of a snake as it is designed to get around the bend without damaging the finish of your toilet.
If you want to confirm its a draining issue, try pouring a bucket of water into the toilet. It should drain almost instantly without the level increasing. If it doesn't, it's clogged.
If the proper amount of water is not delivered to the toilet, it can splash. I experienced splashing of water with an Eljer toilet when the rubber fill hose was not properly clipped into the (vertical) overflow tube. Also, make sure that the new valve is adjusted to close at the proper tank water level.
Another possibility is that the drain is blocked. If this is the case, the drain needs to be cleared. There are special toilet augers designed not to scratch your toilet's finish, though sometimes it may be easier to remove the toilet, clean the drain, replace the wax gasket, and reinstall the toilet.
I had a similar problem of water splashing out of the toilet bowl when it was flushed. This began to happen after I replaced the fill valve. Based on the information here I checked the placement of the refill tube coming off the fill valve. The refill tube had come loose from the overflow tube and just hung down into the water of the tank. I reconnected it to the top of the overflow tube (in my case there is an obvious place for it on the rim of the overflow tube although sometimes there is a plastic clip to hold the refill tube in place over the overflow tube) and the water in the bowl once more flows smoothly without splashing. Thanks for the question and responses.