The size of engine will depend on the aircraft you have. There are two main types of engine used, 2-stroke and 4-stroke. Whilst there are many petrol-fuelled model engines available, these are for models much larger than the basic trainer. The smaller sizes of 2 and 4-stroke engine are generally known as "glow" or "nitro" engines. Their fuel is made from methanol (the same volatile fluid that powers Indy cars and drag racers!) and oil. For extra performance, a small percentage of nitromethane may be added, hence the terminology "nitro engines".
2-stroke glow-plug engines are suitable for all types of aircraft. Which make you decide to buy is up to you but you would be well advised to ask around at the club before buying a particular engine. Some seem to run and handle extremely well, others can be a problem. The last thing you want whilst training is to have no confidence in your aircraft ... and the engine is sometimes the most uncooperative and cantankerous piece of equipment that you have in it!
Everyone has horror stories regarding engines. Listen to the advice and then make your choice. The same goes for fuel. Some engines will only run on expensive fuel, making their low cost somewhat irrelevant. Others will run on nearly anything!
Noise is also important. The engine needs a good silencer so, again, take advice because we do not want to annoy the local residents and loose our flying field.
4-stroke engines tend to be heavier and more expensive than 2-strokes. They do tend to be quieter but can exhibit the same behavioural problems as 2-strokes. For a beginner it is probably wise to leave these to those with some years' experience.
Electric flight is becoming more popular nowadays but the beginner would probably be better advised to wait until they can fly before trying these out.
Minature gas-turbine engines are now also available but these are exceedingly expensive and the 'planes tend to be very quick. Even full-size pilots learn to fly on propeller aircraft before they move onto jets so these are most definitely only for the experts!!!
Take advice from your local model shop and from the club members. Engines are one of the main reasons for failure and frustration when learning to fly. A good engine is worth all the money you paid for it. A bad one is just that and will never perform correctly. It is probably the second most important decision after the aircraft that you will make.