If you liked the 308win and didnt mind the recoil and it appeals to you just get one. You dont need to learn to shoot with a 22LR, but if you want a 22LR then get a 22Lr as well, lots of fun. You may find you progress or evolve your interests over time but really its just a matter of buying the gun you "sort of like the look of, can handle the game you like", and going from there. 308 will cover many things.
basic breakdown for non rifle aware people are as follows
Varmint guns -small calibre very high velocity, good for shooting small stuff to hundreds of metres. With the right bullet and expertise you can also take larger game but need to be careful with bullet choice and picking your shot. eg 222, 223, 22-250
Small/medium bores- As above, the smaller ones in this category may handle long range varminting and they are also more able to take medium game as well. 243win, 6mm, 257,
Medium calibres- for medium game basically. This is what you do most of your shooting goats, pigs, deer etc with. they can also handle smaller game to a degree and big game with careful bullet choice. 30-30, 270, 7mm, 308win, 30-06, 8mm, 303
Big game calibres- Designed for shooting large game( big deer, scrub bulls, camels, horse etc). 338 win mag, 35 whelen, 9.3 x 62
Safari rifles/elephant guns. not where you want to start right now
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What else might be useful for a beginner amid the reams of info...
-At every level calibres can have greater or lesser powder capacity than others . Dont just go off the calibre number, look at the cartridge as a whole.. Eg a 357 magnum is a pistol cartridge designed for small/medium game to 100m, whereas as 35 whelen, same calibre has a much fatter case behind it, burns 3x the powder at higher pressures, and can shoot large game to 200.
everyone has there own idea of definitions and the above is highly flexible, Back to the 308win, its sits around medium calibre level, can shoot some smaller stuff though its a bit overkill at that level, awesome gun on regular game and can take big game with the right bullets.