Here I will go through the steps I use when deciding on which exoplanet to observe using an amateur telescope.
Step 1: Find objects which are bright enough.
Go to the exoplanets.org/table and from the drop down menu on the upper left choose transit planets. Then press the big plus button (+) on the right and choose V mag under stellar params. You can move this table header next to the exoplanet name on the left side for convenience. Click the V mag tab to sort the magnitudes. To only display the transiting exoplanets where the host star has a know V magnitude write TRANSIT == 1 and V in the filter line. The table should now look something like this:
Step 2: Get the ephemeris
Now that you have an idea of which objects are bright enough you want to know when the transit will occur. To find this information go here. On the left you will se the names of the exoplanets. On the right you can click on ephemeris to get the transit times and dates. If you click on the ephemeris for one of the exoplanets you will be presented with a long list of numbers. The left most column is the date and times for when the transit begins. The middle column represents mid transit and the column on the right represents end of transit. At the top you have the transit duration. Remember that all times are given in UT. In this list of dates and times check to see when the next transit will be.
Step 3: Find out if the transit will be visible from your location
Copy the name of the star which the exoplanet is orbiting, like HD209458 and do a search on SIMBAD to find the coordinates. The name might change. The coordinates you are after are the ICRS coordinates. In this case it would be 22 03 10.78 +18 53 03.7 I would recommend you check the coordinates with those listed on the ephemeris page.
Now that you have the coordinates you want to see when the object is up in the sky (if at all). What a lot of professional astronomers use for this is staralt. Start of by choosing the date of which the transit will occur. Next it is unlikely you are at a professional observatory so here you will have to enter the coordinates of your observing location. A nice and easy way to get these coordinates is using google maps. Navigate to your location on the map, right click and choose directions from here from the drop down menu. That should give you the coordinates in the text box found at the upper left of your screen. As an example of using staralt I will choose the Norman Lockyer Observatory, in the southern UK which I found (using google earth) has the coordinates 50.687901, -3.219783. The observatory is at a height of about 100 meters above sea level so in the observatory text box in staralt I write: 357.00 50.687901 100 The staralt page should look something like this:
Hit retrieve and a graph will appear. On the left y-axis we have elevation whilst on the x-axis we have time in UT. The dotted vertical lines represents astronomical twilight whilst the curved dotted line (if there is one), represents the moon. An ideal target will have a parabola shaped curve peaking at sometime during the night. Here is an example of an output showing two example targets:
Target1 22 00 00.0 +19 00 00.0
Target2 12 00 00.0 +19 00 00.0
Target1 is seen on the left and it shows us that is is rising during the morning ours. This object would not be very suitable as I would not like to stay awake that long and also once it rises above 30 degrees twilight has begun. Pretty neat huh?
Target2 is on it’s way up in the beginning of the night peaking at a round 22 UT. The object is visible for quite a while not setting before 4am. A target with such a visibility curve would be ideal.
So there you have it, repeat the 3 steps until you have a suitable target for the night you wish to observe. Keep in mind, if you are looking for transiting exoplanets with a host star brighter than about 10th magnitude you will definitely not have a transit happening every night.
If something was unclear in my description please comment below and I will answer the question. If you find that something is not clear I am sure other people think the same and will be glad you asked.