To add to @
FireOfExile
's post:
Little Mac, Palutena, and the Mii Swordfighter also have counters.
Grabs default to the Z buttons on the Gamepad, classic controller, and Gamecube controller. They can also be done by shielding and pressing the attack button while shielding (in fact, the Grab input is, from a technical standpoint, just a simultaneous shield and attack input). Once you've grabbed an opponent, you can keep hitting the Grab button or Attack button to pummel them, which increases their damage. However, your opponent is also able to struggle out of a grab by more or less mashing buttons and directions. It's important to use the throw (via pressing a direction) before the opponent breaks out.
Furthermore, there are a few characters who have what are called "command throws". These are basically special moves that function like grabs, which is to say, they go through shields. The ones that spring to mind are Kirby and Dedede's Inhale, Robin's Nosferatu, and Diddy's Monkey Flip.
Improvement is best attained through practice, in my opinion. If you have siblings/friends to play with, especially ones about the same level, using them is the best way to improve. Of course, you can also play online, and either For Fun or For Glory should give you practice against a variety of playstyles and characters. Single player modes can also help you develop a grasp of all of the game's fundamentals (shielding, dodging, rolling), as can Event Mode in some cases. Basically, use everything at your disposal.
Personally, Training Mode is less intended for learning in general, and more for refining techniques. It's good for finding out specific damage and knockback values, and at what threshold certain attacks will kill enemies.
Once you develop a moderate grasp of the fundamentals, consider looking for guides, combo videos, and professional players playing the characters you like. Doing so off the bat can be good if you're a very quick learner, but jumping straight into the advanced without a firm grasp of the basic is a quick path to frustration, in my experience.
Hope that helps! Welcome to the Boards.