The name of 'Ayin means 'eye', and if you squint a little you might see it as a picture of an eye. Many speakers treat 'Ayin as a silent letter like Aleph, but its original pronunciation was a glottal stop.
You make a glottal stop by contracting your throat and stopping your breath for a moment, almost as if you're going to choke. The sound in the middle of the negative expression "unh-uh" is the closest thing we have to a glottal stop in English.
In the towns of Europe, the art of the glottal stop was lost. (In fact, the letter 'Ayin was re-purposed in the Yiddish language to represent the vowel E, so that the Yiddish word for 'better' - [beser] - is written [בעסער].) But the Jews of the Middle Eastern world lived surrounded by Arabs who'd been glottal-stopping all their lives, and so the sound lives on in Mizrahi Hebrew in all its full-throated glory.
In practical terms, 'Ayin is the second of those two guttural letters (after Chet, remember), and when it comes at the end of a word your mouth needs an extra moment to slide into position. This is where the 'furtive Patach' comes in. When a word ends with a Chet or an ‘Ayin, we insert an “ah” vowel sound before the final letter - but the vowel (patach) is written under the letter, as if it came after the consonant. So we have:
לוּחַ - pronounced luach (not *lucha) - table (chart)
רֵעַ - pronounced rea’ (not *re’a) - neighbor
The name of the letter Peh means mouth. With the Dagesh it has the sound of P, and without the Dagesh it sounds like F. Normally, Peh will always say P at the beginning of the word and F at the end; the only exceptions are a few newfangled or foreign words like 'lap-top' (just what it sounds like), 'fisfes' (he missed or failed to hit), and 'frecha' (a bimbo). For those special cases where we need to show that it has the F sound instead of P, we sometimes write it with a bar or long mark across the top.
The Peh has separate regular and final forms. The cursive Peh looks almost like an upside-down version of the printed letter, and the cursive Peh Sofit looks a bit like a musical G-clef.
The name of the Tsadi might have meant "hook" in ancient times - nobody is really sure - and in modern times it is sometimes also called 'Tsadik', which means a righteous or saintly person. It is generally pronounced as the 'ts' consonant combination. Written with a Geresh [צ׳], it makes the 'ch' sound in foreign words like 'chat' and 'ranch'.
Write the printed form of the regular Tsadi carefully so that it doesn't look like an 'Ayin. The printed Tsadi Sofit has a long vertical stroke that hangs below the line. The cursive Tsadi looks a lot like the numeral 3; the cursive Tsadi Sofit looks a bit like a cursive capital "L" - notice that it doesn't take an extra loop like the Peh Sofit.
The name of this letter is either Kof or Kuf - take your pick - and nobody knows what it means. It makes the sound of K. Some speakers distinguish the sound of Kuf from the sound of hard Kaf: Kuf gets the far-back-in-your-throat K sound, while Kaf gets the roof-of-your-mouth K sound. (And for this reason, the letter Kuf is sometimes transliterated with a Q; you'll sometimes see it this way in words like 'Qabbala'.) But most people pronounce Kuf the same as hard Kaf.
Kuf is the only other letter besides Heh that's written as two separate strokes. Sometimes people who are extra-careful about respecting the Divine Name will substitute Kuf for Heh in the name 'Elohim' (G-d) so that it's spelled and pronounced 'Elokim'.
Kuf doesn't have any alternate forms to worry about, and the cursive form looks pretty much the same as the printed form. It's interesting to note that Kuf is the only letter, other than the final forms, that hangs below the line.
The name of Resh means 'head' in Aramaic (in Hebrew it's 'rosh') and it has the sound of R. Israelis generally pronounce their R's back in the throat, like the French or German R; if you master this sound, you'll improve your accent and sound less like an "Anglo-Saxon" (i.e. a newbie from North America).
The Resh looks mostly the same in its printed and cursive forms. When printing, make sure you draw it with a rounded corner so it doesn't look like a Dalet.
SHIN / SIN
The name of the letter Shin means tooth. This letter is a special case: it's written with a dot on top, and when the dot is on the right, the letter has the sound of 'sh'; with the dot on the left, it says S.
The letter Shin sometimes appears on ritual objects (mezuzot, tefillin) and is said to stand for Shaddai, one of the names of G-d.
The name of the letter Tav means a sign or seal.
In pronunciation, Tav is a unique case. The soft Tav is the main shibboleth of the pronunciation differences between Ashkenazi and Israeli Hebrew: Ashkenazim pronounce it as 's' while Israelis say it as a 't'. In ancient times it probably had the sound of 'th' and was interpreted differently by different populations in later years. (I think of the New Yorker who says "Whaddaya tink?" while the German says "Vat do you sink?")
In Hebrew words that passed into English in early times - usually by way of Latin or Greek - the 'th' spelling was preserved. Hence: English 'Sabbath', Ashkenazi 'Shabbos', Israeli 'Shabbat'.