Welcome! Shimmerfly's link is full of info you'll need time to absorb; I still go back and read it. But, I wanted to give you an answer in this thread specific to training size and squad size.
Training size is determined by the number and level of armories (as well as some Ancient Wonders later in the game). Squad size is determined by the number of 'Squad Size Upgrade' techs you have activated in the research tree. If you open your barracks and go to 'Army Camp' there will be a number in the top right corner that shows your current squad size. So, if your squad size is 200 but your training size is 50, it will take 4 of the training queues to get an additional squad added. Also, when you activate a new 'Squad Size Upgrade' tech, your number of squads will go down as it takes more units to make up a squad. But, since there are more units to the squad, it is stronger.
A higher Squad Size will make battles on world map provinces easier. Also, the early chapters are where battles are the most difficult (no, I don't know why; makes no sense to me
) so even SS upgrade techs may not make much of a difference in the difficulty level of the fights. This will improve in later chapters.
A higher Squad Size will make tournament provinces more expensive in both troops and goods. The enemy squad size in tournaments is based on your squad size. So, if you expect (pulling these numbers out of the ether) to lose 10% of your troops even with a win, you will lose 10 troops with a Squad Size of 100, 20 troops with a Squad Size of 200, etc that you will need to train more units to replace. There are optional Squad Size Upgrade techs in the research tree. Optional meaning they don't open more techs in the tree when completed. General consensus is to start skipping the optional ones once you reach Ch 5-6. Tournaments occur weekly and you'll do way more tournament provinces than you will map ones (or you should, anyway. Tourneys have lots of rewards!) so it's better to skip the optional ones.