I'm going to skip over the Chat part of your question, as I've never bothered to try the chat here. However...
First of all, what you see by way of smilies will depend on what kind of text box you're using. Those little "quick reply" ones offer the fewest features so I usually click the "Go advanced" option.

In the "Go advanced" mode there should be a box at the right showing 15 of the most common smilies. If you click on one (left click, as Babysteps says) it should insert a smiley (or the text that represents it) where you were just typing.

At the bottom of that box of smilies is a link to "More". If you click that, a new window (little at first) opens up in the corner and you can scroll around in it to find more smilies than you can shake a stick at. It's easier to see what-all is available if you maximize that window while making your choice.

In that smilies window there are two representations for each smiley -- its picture to the left, its text to the right. Clicking on the picture inserts the smiley where you're typing -- most of the time. Sometimes nothing happens and I have to click a second time. Clicking on the text string doesn't do anything but if you type (or copy-paste) that text into your post like this :example: you'll also get that smiley. As far as I know, there's no smiley called "example" but if you substitute "hug" (with the same punctuation) you get

Try it and let us know how it works. If you have any difficulty, do let us know exactly what steps you follow to try to get a smiley, what you actually see, and what happens after each step.