I assume that because you are asking about champion pools, you want to rank up & improve. If you are only looking to have fun, then just play whatever you want. It's really good to write down your champ pool to remind you to stick to it.

Try to narrow down a champion pool. Start with a core of just 2-3 champions for the roles you like, and then expand a little when you have plateaued a bit (like when you stop improving quickly). Getting good with a limited number of champions means you can learn every match up for your main few picks, and you should be able to climb more effectively.

You do want some balance in your pool, for example, top lane you might want a mix of tanks and carry tops, and then later on you can add like a mage top sort of thing. However i dont think this is as important as just carrying on with a small number of champions. If you never want to play tanks top, you don't have to if you only solo queue. A team might want you to play a better mix of champions though if that's what you are looking for.

Don't focus in too hard on counterpicks. I'd rather play a horrible match up for one of my main champions, than play a nice matchup for something I am inexperienced with. If you have 100+ games with a champion, and someone tries to pick a counter to it that they are not familiar with, most likely you will still stomp them.

You also want a balance in terms of champions you can play well with, and champions that are fun. If a champion is really easy to master and climb with, but you have no fun on it, you are not going to be motivated to play it. However if a champion is super fun, but its too hard for the player in question, for example lee sin in low elo, you might be motivated to play it, but you are seriously holding yourself back by taking it.

It's nice to take champions that can be played in your other roles. For example, with top/jungle, you can take certain champions to both roles, and it will help you learn faster, like trundle, malphite ect.

Support champion pools are similar. My ideal support buddy would have 3 champions, and they would fill different roles, like one engage support, one high utility with peel and shields/heal, and one damage support. But if you really don't like playing damage support or utility, and just want engage, then taking 2 similar champs is a way of quickly improving your engage supporting.

Personally, I want to be versatile whilst playing as few champs as possible. I have decided on just lucian, vayne & caitlyn for adc for season 6. They all provide me with something different. When i reach an elo I am more happy with, I will add another ad. Similarly for mid, I will take Ahri, diana, lulu. If I have to top for some reason, I would only take Irelia or wukong, and later on in my climb I might try to add malphite and replace one of my first 2 with fiora.

Just examples of course, good luck!