Breeding animals can help. How quickly you gain xp depends on the number of animals you have, and the more animals you have, the more crops you will need to feed them. It will also stock up your food supplies, and cooking the meat from the animals will give you a little extra xp from the furnace. But the xp from cooking the meat is pretty small.

If you try breeding animals for xp, I'd suggest building up your population until you have at least 20 adults for each animal you want to breed. After that, you can start killing the mature animals to keep the population constant.

Regardless, getting xp is meant to be time consuming. There is no "quick" way about it without exploiting some of the game mechanics (i.e. mob farms).

That said, a mob farm does not require a spawner. The simplest mob farm is a dark room. The simplest would be a large, dark room with a door. You enter, kill everything manually, leave, move at least 24 blocks from the room, wait some time to let mobs respawn, go back in swinging, and repeat. Some designs use dispensers with water buckets to periodically flush water that pushes the mobs inside to a central location for easy killing (without having to enter the room). Another design by popular youtuber Etho relies on the mobs wandering into a water stream that pushes them where the player wants them.

These types of mob farms are not complicated, but at the same time, they are not very efficient (i.e. fast). If you want to ease yourself into the idea of building mob farms, I'd try one of these first.