I'm sharing this tip assuming that you have at least a few grain nodes(barley, wheat, potato, corn), some ore nodes(iron, copper), and some wood nodes(birch, ash, etc) because this is the easiest way to level up processing without spending any money, However, if all you care about is the exp, then go straight to the second portion.
1.) Making money while you level processing
This is why you'll need some grain and ore nodes, because you'll only be processing what your workers make for you, it's going to start out slow if you don't have a good worker setup or if you didn't take these nodes (which you really should they're the best production nodes in the game).
So if you're going to have a long AFK session, just grab as many grains as you can and grind them into flour, the grains have a 1:1 ratio, so 1 grain will give you 1 grain flour. Meaning this will take awhile to actually finish, however the flour costs more than the grain on the MP and so as I promised, you're making some money while you level up your processing, do take note that processing grain into flour only gives you 75 EXP, so only do this if you're going to AFK for a long time, and let your primary source of exp come from turning ores into melted shards, and wood into planks, which give 200 EXP each.
Of course you can take it one step further and process the T2 mats you just obtained into T3 mats and get even more EXP, but I'd suggest waiting til you get a higher processing level like Artisan 2, because once you hit Artisan 2, congratulations you just hit the processing soft cap, you now average 2.5 procs for every material you process.
2.) Power leveling with no care in the world
The easiest way to power level processing? Flax, and cotton. These weigh very little and can give you quite a good AFK session, and not only that, but everytime you successfully process flax or cotton, you get 500 processing EXP for just getting the T2 material. For comparison, processing ores into melted shards, and wood into planks will only give you 200 EXP. But that's not all, after you process the Flax threads or cotton yarn into fabric, you'll get 1000 EXP, which is obviously a lot and still weighs a lot less than Ore/Wood.
However, the downside to this is that, obviously you had to spend silver to buy all the materials, and cotton and flax byproducts barely sell on the marketplace, you could have cotton/flax fabric sitting on the MP for days without anyone even touching it. You could open up a clothes workshop to produce life clothes and actually use the fabric you produced to make some money back.
Thank you for reading, and good luck :)