Assuming you have a budget in mind, the first thing you need to do is find out how big your lawn is. You can either find your lot size on your memorandum deeds if you've bought your house, or you can check your city's website to see if you can request lot measurements if you're renting. If your lot measures about an acre, you'll be able to use a 30 or 42-inch cutting deck without any issues. For lawns up to two acres, a 42 or 46-inch deck is ideal. And if your lot is over two acres, you can get a mower with up to a 72-inch cutting deck to handle larger areas.
The transmission type is also important. Many newer models have what is known as a hydrostatic drive. This means that they operate similarly to how a car drives: You push the pedal and it moves forward or backward. And the harder you push, the faster you go. This makes it easier to learn how to drive, but that also makes the mower more expensive. More stripped-back models have variable speed manual transmissions, which allow you to "set and forget" your speed so you can focus on paying attention to obstacles and people who may be nearby.
Also: How to choose a lawn mower
And finally, you'll want to consider the power source for your new riding mower. Gasoline engines are far more common, but there is now a wider variety of battery-powered models to choose from. The perks of a gas engine are that you'll get near-infinite run times (as long as you have enough fuel to keep the engine going) and a bit more power for handling steep inclines and rough terrain. The downsides are dealing with exhaust emissions and maintenance that can be a time and money sink. Electric models don't need engine maintenance, so you save a bit of money in the long run. But they usually have a maximum run time of about an hour, which means that you may have to plan your mowing over several days if you have a larger yard.