When we think of astronomy we almost always think of telescopes but astronomy can be appreciated and enjoyed without a telescope so if you are interested in astronomy and don't have a telescope don't let it dissuade you! There are lots of interesting things to see and enjoy in the night sky with just your eyes.
Some Important things
The effect of Light Pollution: The most important thing to think about when viewing the night sky without a telescope is that light pollution is your biggest enemy. If you live in the city or near a source of bright lights you are going to get a significantly reduced view of the sky. If possible you need to get away from the city and any bright lights. Take a short road trip out to somewhere less populated. It will make a big difference in what you can see and instead of seeing hundreds of stars you will see thousands! It really makes a big difference.
Let your eyes adjust: Everybody is familiar with how your eyes dilate and open up in darkness and this opening up is important. But you need to realize that your eyes will not make their full adjustment in a few seconds. It takes a few minutes for your eyes to relax and open up to their full light gathering capability. So, when you are intending to view the night sky you should go out into the darkness and realize that you need a good fifteen minutes for your eyes to reach their peak. If you go in and out of the house this effect is ruined and you need to readjust.
About flashlights: If you are using starmaps or starcharts you are going to need some kind of a flashlight to look at them. Using a bright flashlight can ruin your night vision so what you need to do is tamper down the brightness of the light with some kind of a cover. use a red or blue transparent shield over the flashlight. Your eyes are not very responsive to red or blue light and your pupils will not dilate under a mild red or blue light. This is a trick that the military has been using for a long time and you can buy flashlights with colored lenses or even make one your self.
About Starmaps and accessories: The night sky is chock full of all kinds of interesting stuff and while you can enjoy exploring it on your own it will be a much more rewarding experience if you have some help in the form of star charts and other materials.
About the Weather: This is something you really should consider. If you don't spend a lot of time outside at night you might be surprised by the weather conditions. If you live in a climate that has a winter season it could get extremely cold outside at night. Even in warm climates the temperature can drop significantly at night. So, if you are traveling off to find a dark place to do some astronomy remember to bring plenty of extra warm clothes with you, maybe even a blanket and hot chocolate.
You might want to consider this book. If you want to adventure into astronomy without yet spending the money on a telescope this is a great way to begin and as your love of the hobby grows you will want to buy a telescope in the future.
Let's take a look at some of the great things we can see in the night sky without the aid of a telescope.
The Moon: This is the first and most dramatic thing you can look at in the night sky. And you might not have considered it but it is also a big source of light pollution so if there is a full moon you probably don't want to try observing deep sky objects. And the moon itself is best viewed as a crescent. The crescent form gives the longest shadows and the best contrast in view. You see much more detail. Here is a great interactive map of the moon
Stars: Now before you laugh because this is quite an obvious statement you have to give a bit of consideration to the fact that every star is different! Under a very dark sky and after your eyes are adjusted you are going to realize that stars come in a remarkable range of colors and brightnessess! Just sit back and soak it in.
Planets: Yes, you can see planets in the night sky. But they move in relation to the rest of the sky so you will need a planet locator chart to identify them. A good rule of thumb about identifying planets is that they don't twinkle! Stars often twinkle and this is due to atmospheric disturbances but planets do not twinkle! If it doesn't twinkle its probably a planet. Mercury, Venus, Mars, Jupiter, and Saturn are all visible with the naked eye at various times of the year. Here is a great tool that will help you find the location of the planets
Constellations: These are groupings of stars that form patterns people have interpreted as heroes, animals and other objects. Learning the constellations and the history/name of each is a lot of fun. It also helps you to identify other objects in the sky. It gives you a good reference to what is where.