Setting up is really simple. At the minimum, you need is any web browser. However, to get the most out of this hobby, you may want to look into using some of the following:
Ideally, the best thing to use is a computer, since then you could keep your log, data, and information on the computer. Also computer-based decoders and recorders may be easier to use than mobile apps. However, there are many mobile apps that you can use to get started and if all you're doing is listening but not keeping track, a mobile device will be all you really need (along with a web browser app, headphones and optionally a pen and log book or journal).
This can be any browser you are comfortable with. Be sure your web browser can support Java, Flash and/or HTML5 since most webSDR sites rely on one or more of these requirements. See the Resources section for a list of webSDR sites or just search in Google.
While you could listen through your computer or device speakers, often times some signals can be rather hard to hear. Using headphones can help you hear things you wouldn't ordinarily be able to pick out. Then you could fine-tune the radio to get a better signal. I use over-ear earbuds as that is all I have right now. Ideally, using noise-cancelling headphones is probably the best.
Keep a pencil and journal handy to write down anything you find interesting. Keep record of time, date, WebSDR URL, radio information (which can usually be found on the web page or one of the SDR's tabs) such as city/state, the radios call sign, band/mode (ie. 160m LSB for example), Frequency (7065.75 or 7.065MHz for example), if there was a tag at the top of the waterfall, what it said for that area, what you heard, what you decoded (if applicable) and duration of transmission. Some may even like to use a spreadsheet to record this information. Keeping a log may help you identify certain stations you might want to revisit, and what is on them (or alternatively, stations you want to stay away from for whatever reason).
Whether or not you can record these transmissions depends on the country, state and/or local laws. Be sure you are allowed to record. Some WebSDRs have a way to record what you hear and save the recording. You may also be able to capture the sound on your computer or phone. This is useful for decoding Morse Code, for example. See Captring Sound and Recording for more information.
If you want to decode Morse Code (or other signals) you will either need to learn Morse Code or you will need a decoder. There are many kinds out there. See Setting Up A Decoder for more information.
Listening takes time. You may not get much at first, and may have to look throught he frequencies for awhile to find something that interests you enough to pay more attention to. Don't give up too fast. Some nights you could go a couple hours (or all night) and not find anything. Other nights might be more active. Remember, it's a fun hobby and like an adventure. Take maybe an hour or two after work, in the evenings, or on weekends whenever you have spare time for hobbies and recreation. This is one hobby that does take some patience and perceverence.
Computers and mobile devices, as well as headphones can pick up intereference which may cause problems. If you are experiencing interference, be sure to check the following:
You are away from anything that can radiate interfering signals, such as WiFi routers, AM/FM or other radios, Fans, Air conditioners, Refrigerators, some types of ballasted lamps, other mobile devices and computers.
You are not near any TV and other antennas that are in use. Even headphones can pick up signals as some mobile apps require headphones since they would use the headphone cable as an antenna.
Check your headphones to be sure the cable is sheilded enough not to pick up extra noise.
Check your computer (or device) sound mixer settings to be sure that you do not have a microphone on that may be feeding ambient room noise to the speaker or headphone jack. Many computers have a sound mixer program that lets you control (or even turn off) the microphone.
If you suffer from Tinnitus or other type of hearing loss, it may be hard to distinguish some signals from the ringing in your own ears or you may not be able to hear everything. Do not be discouraged. If you find your own condition to be limiting in some ways, you may want to stick to listening for only certain signals, or listening to audio (talk) communications only. Using the waterfall on the WebSDRs can be useful though in narrowing down signals as they show a visual of how strong a signal is. You could rely on this as a guide as well in picking up other signals.