There are several million words on the internet written about beer tasting (yes, I counted them). You can find advice both good and truly horrible, ranging from the simple admonitions, like "put it in your mouth," all the way through to extremely pretentious suggestions that would put a wine sommelier to shame. You can find color charts with more choices than Sherwin-Williams and style categories so narrow you can't shove a dime into them, much less find three beers that fit.
That said, though, there are some key points that I think will make your beer tasting just a bit more thoughtful. Please don't, however, let "tasting" ever get in the way of "enjoying," which is really the point after all. Pretension doesn't go well with any beer.
Color - The first thing to do with any beer is look at it. Seems obvious, but a lot of time we pick up a glass and put it to our lips without a second thought. To get a better sense of the beer, just take a look. This is usually relatively easy, as most places will serve you a beer in a clear pint glass, but take a moment to pick up the glass, maybe hold it up to a light source and then consider the color.
The color may range from very light straw to a very dark brown or opaque black, but if light does transmit through the beer, the next step is to consider how cloudy or clear the beer is. The amount or type of filtering differs between styles and breweries, but take a look for a consistent texture through the glass. If, by the way, you find oddities floating in the beer or flakes/chunks drifting down to the bottom of the glass - that is a bad sign, although usually that is from the line from the keg to the faucet. You might ask about the tap line cleaning, or you might want to find another establishment, but in either case, keep in mind that flakes shouldn't be there and it may impact the flavor of the beer.
Smell/Aroma - After you look at your beer, the next step is smelling your beer. This is the stage where a well-poured beer, which has a head on it, will enhance the opportunity to smell the beer. A flat beer or one with little or no head will have less aroma available for your nose. A decent head on the beer will help aerate some of the aroma, making it easier to get a sense of the beer. Beer is made up of two primary ingredients - malted barley and hops. Usually one or the other will be the predominant smell, i.e., hops stand out with an IPA, while malts will be stronger in most stouts or porters, but the smell of both hops and malt are discernible in most beers.
Some glassware is better for smelling (and this is why most wine glasses are curved inward at the top and wine drinkers will stick their noses into the glass), while the most common pint glass still is not really well suited to gathering smells, but still put your beak down to the head or liquid and inhale. It's worth it.
Mouthfeel - Next stop on this sensory experience - put some beer in your mouth. Good step, eh? Also, don't sip, slurp or suck beer in. The ironic thing about people who are hesitant to drink a new beer (picture the macro-drinker asked to sample a craft beer) is that the tiny sip (followed immediately by a declaration said craft beer is "bitter") is that this process aerates the beer in a way that is intended and strongly enhances the bitter flavors. Give it a try sometime - bitter abounds. That's why beer tasting is done with a "drink" rather than a sip.
When the beer is first in your mouth, consider for a moment how it feels. Beer can be thin or thick feeling in your mouth; very carbonated or not (this may depend on the gas pressure or gas mix to pour the beer); very viscous or very light.
The term "mouthfeel" usually gets some laughter when I mention it at a tasting, because it is a funny word and not a familiar word, but if you think about mouthfeel while the beer is in your mouth, you will be surprised to notice how distinct the mouthfeel is of many beers.
Taste - Next up, the most common step and one that is almost unavoidable - taste the beer. The taste of the beer is the whole mixture of hops, malt and anything else in the beer, combined with the carbonation level, the gas used to pour the beer (usually CO2, but can be nitrogen or air, or a mixture of these, purposely or not), the temperature of the beer (same beer at different temperatures tastes very different) and the combination of other things that were in your mouth just prior to drinking this beer (whether food or a prior beer) and sometimes the smells or atmosphere around you.
If you are really wanting to taste a beer, it is better to eat something very neutral (mild bread or crackers) to help clear out your palate, but even if you don't do that, be aware that your mouth is a dynamic place that mixes up whatever you have going on, so at least make mental note of that.
If you are going to be tasting numerous beers, also keep in mind that your best experiences are going to come from drinking the lightest beers first and working towards the darkest and/or hoppiest. Hops give their flavor from the hop oil that is in the beer and some of that oil will linger on your tongue past the drink, so the more hop in your beer, the more impacted later beers will be.
After-taste - And lastly, the after-taste. After you swallow the beer and consider the tastes that were present in the beer, then wait a moment and consider what flavors are left over. Flavors that linger in your mouth or on your tongue tend to become sharper or less appealing, so after-taste can have a big impact on what you think about a beer. For this reason, the less after-taste the better usually, but in any case, take a moment to consider it before you move onto the next drink.
Enjoying beer is really the reason we drink it, but taking a moment to thoughtfully drink a beer can increase your enjoyment.