This all comes down to faith as some people don't believe in them, but yes they are like spirits. You cannot turn into one or 'gain the powers' of a fae but you can call upon them in your spells to help with wealth, protection, love and so on. Different faeries have different strengths and weaknesses [like a earth faerie or a water faerie/sprite] and you should befriend them before demanding favours, but you can work with them in your practice.
To attract faeries you should plant a faerie garden with things they like such as sweet smelling flowers [ones that attract butterflies and hummingbirds] saffron is a favoured plant. If you plant fruit trees, leave some fruit behind as gifts and faeries don't like greed [they have been known to bless or curse people they like or don't like] some people can see faeries, but most people I speak with see them as orbs [a vibrant purple but that could just be the ones in my area]
You can not catch a faerie, and you can not "take" their powers. They generally avoid people with such thoughts as it is anyways. I'm sure you could attempt to enact something that you believe would catch one, and upon it having no success, you would probably find yourself with a cacophony of bad luck. Simply because the Fae can cause you bad luck, and because you tried to cause them bad luck, they would love to return the favor.
However, you can form a friendly relationship with them if they happen to dwell or pass by your place of abode. Leaving out a dish of milk and some bread is a good way to show your good will, and like Neko said, you could plant a garden to attract them. But they are beings of vibrational energy to us, and they would sense your ill will in a heart beat. Small gifts are good invitations as well. If you've ever been walking in the woods, and found something like a rock, or piece of bark, or flower that caught your eye for some reason, you could always offer it as a gift to them. Some of them delight in the beauty of what we call simplicity, they see it in a different way.
I've read about people saying they have seen the Fae, however it was unclear if that was in their minds eye or if they actually saw them. On this subject, I always play it safe and assume they meant within the minds eye.
I love the discussions on these "nature spirits", now to answer your questions, it depends on if you are talking about little people with wings or the spiritual belief of the fae, none of them exist on this plain, so no you can not physically see them.
Fairies can be seen, as they can manifest themselves the same way ghosts or similar spirits do. In my home, I will encounter many beings, but there is always a small flash of light I can barely see. What makes these lights different from the other spirits is that my youngest brother can see them too, and i'm the only one who has seen anything supernatural in the house. I am certain they are fairies as when I tried to make contact them, Calling them as fairies, I saw a small shadow out of the corner of my eye dart behind one of the couches. I took it as a sign from them.