Check the official Unicode Consortium website. Their charts list every Unicode character’s unique hexadecimal value. ( http://www.unicode.org/charts/ ).
You can create an HTML hexadecimal entity from each code. You enter “& # x” (without the quotation marks or spaces), follow it by the code (in upper or lower case and omitting leading zeros if you want), and follow that with a semi-colon (;). For example, the first symbol on the chart you provided has the hexadecimal value 2600. So in many forums, including this one, if you enter “& # x 2 6 0 0 ;” (without the quotation marks or spaces), when you preview or post, the HTML hexadecimal entity will turn into ☀.
If you are using Windows, I recommend you download and install the free Quick Unicode Input Tool from http://www.cardbox.com/quick.htm which enables you to enter any of those symbols by holding down the left Alt key, pressing the dot on the numeric keypad, typing the unique hexadecimal number on the keyboard or numeric keypad, and then releasing the left Alt key. (The right Alt key also works on many systems.)
If you are using a notepad computer, you will have to emulate the numeric keypad dot. See the article “Fn key” in Wikipedia for one method.
Note that while Quick Unicode Input Tool is running, the regular DOS Alt-code method for symbols like ☺ and ♥ will not work. You will have to use the longer values from the charts.
In Wordpad on Windows machines and in Microsoft Word, if you type one of those numbers into your text and follow it with pressing down the left Alt key and then pressing x, and then releasing the Alt key, the preceding number will change into the corresponding character. (The right Alt key will also work on many machines.)
For further information, including how to use those values on the Macintosh and in Linux, see http://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&item_id=inputtoollinks .
Note although word processors have some degree of automatic font substitution, you mostly must have the font set to one which contains a character in order to see it. You can just look through your fonts at MenuGlobe/Start → All Programs → Accessories → System Tools → Character Map for any character not greater than U+FFFF. Web browsers have a larger degree of automatic font substitution, Firefox appears to be the best at displaying a symbol even if it is only in a single font on your system.
You can paste the character itself, or its unique hexadecimal value, (or its unique decimal value,) or part of its name into the Query box at http://www.fileformat.info/info/unicode/char/search.htm and mouse-press the “Search” button. If you get more than one answer, pick the one you think best and you will get an information page about the character. (Some characters recently added to Unicode do not yet appear in this method.) Try breaking up Ƹ̵̡Ӝ̵̨̄Ʒ into its separate characters and use this site:
· Ƹ · ̵ · ̡ · Ӝ · ̵ · ̄ · ̨ · Ʒ · The center dots are added by me to divide the characters. I have also inserted a space before and after each of the characters.
On the information page, if you click on “Fonts that support ...”, you will get another page that lists some fonts that contain that character. On this other page, if you click on “Local Font List”, you will get a list of every Unicode-compatible font on your system showing how the character looks in that font. For an unusual character, in many of the fonts it will just show as an empty box or bullet or some other symbol which indicates it is not found in that font.
Good fonts for the Latin alphabet that contain lots of symbols and other characters are the free DejaVu fonts from http://dejavu-fonts.org/wiki/index.php?title=Main_Page, the free Stix fonts from http://www.aip.org/stixfonts/font_download.jsp (lots of math symbols), the free Charis SIL and Doulos SIL fonts from http://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&id= , and the free Andron Scriptor Web and free Palemonas MUFI fonts from http://www.mufi.info/fonts/ . For symbols, also download and install the free Symbola font from http://users.teilar.gr/~g1951d/ .
See also http://www.alanwood.net/unicode/fonts.html for more fonts that contain a large number of symbols or characters used in a particular language, most of them free.