One of the hardest parts of learning any language is grasping pronunciation. It's hard to learn pronunciation through transliteration though, so I cannot stress enough the importance of learning to read and use the native writing system, because each character was designed to fit its own sound, making the writing of and therefore memorisation of a language easier. Most of you have probably already memorised your Cyrillic, so I'm not going to focus on mnemonics or anything. Instead, I'm going to talk about writing it (especially in cursive) and making it look believable!
As a lifelong Cyrillic writer (though my repertoire features variations on what you learn in Russian), I am going to give those trying to make their writing purty a tip: invest in a really good fountain pen. If you're trying to get really good at your Cyrillic with a cheap, sticky inked BiC pen, you can probably forget it - the stress on your hand from the friction caused by the viscosity ruins your strokes. Doesn't matter if it's cartridge or converter filled, a fountain pen will really improve the flow of your handwriting. To this day, fountain pens are popular and often preferred writing tools in the eastern hemisphere, especially in Eastern Europe.
Why is this?
The liquid ink of the pen allows it to smoothly glide along the page, helping to produce the long strokes paramount to Cyrillic cursive writing. In this way, it's rather similar to a fine paintbrush (a quality which also lends itself to writing Asian scripts, particularly those with a mass number of strokes). A liquid/gel pen will produce a similar result BUT I find it's too light, while a good fountain pen is also heavy enough to provide balance, allowing it to guide your hand in smooth loops and transitions, as well as keeping the pen at the right angle at almost all times, making for even lettering and slant to your writing.
For example, I have written out this tongue twister "Карл у Клари украл корали, а Клара у Карлa укарала кларнет" in both BiC and fountain pen, and in handwritten and cursive fonts.
First is the BiC, with the straight hand at the top, cursive at the bottom.
You'll notice some jerky corners and hesitation, as well as inconsistencies in the slant.
Next is the fountain pen in the same order
I don't know if you can see all that well in poor scans, but there is far less hesitation and jerky movement in the fountain example. The consistency of the ink's colour also really helps, especially for those of us with problems with low contrast writing (e.g. I can't read pencil).
Further, the nature of a fountain pen nib ensures that you're not going to be pressing, allowing (again) for smoother transitions as well as reduced cramping and tiredness in your hand (and destruction of your page).
Fountain pens needn't be really expensive either. If you don't wanna blow a couple of hundred on some sort of far out Waterman, I recommend Parker's Vector, which I use with a medium nib for all my day-to-day/school needs. It's slim, it's light with a metal end to provide extra balance, and it cost me only $15 CND and is going strong.
When you do actually begin writing on the page, it's often a good idea to just give your hand a wild shake to loosen it up. Don't try to write with your fingers or you'll end up with cramping - smooth hand movements are key.
Don't be afraid to just continue through the word with your loops, making your strokes primarily upwards an downwards to form nice lines. When you do slanted cursive Cyrillic, you may find all the strokes and loops generally look the same and it's hard to distinguish the letters, but as long as you are consistent, the letters should be fairly apparent. Everyone has some sticky letters - you may find you lose concentration and start writing a certain character in Roman alphabet almost all the time, no matter what you try. This is normal, and I suggest simply repeating the writing of different whole words containing this character, instead of just the character itself.
If you find your motions are still jerky, remember to always keep your hand as loose as possible!
For those wanting to type Russian, you've probably found that the default keyboard layout is all kinds of terrible. There is a way to fix it though, you can find it here (see instructions for "phonetic layout").
So that's pretty much it for my preachiness! I would recommend learning the cursive, as it really speeds up your Cyrillic and allows you to have a really awesome autograph XD
HOW TO WRITE CURSIVE
Learning cursive is something that comes with time, and develops as you become faster (and messier) in your writing. As you may notice, regular script has a hard time connecting, which means you can't get really fast flow and writing (it's the best one if you're using a disposable pen). There are some general rules that lend themselves to writing cursive in any script so I'll detail them here.
First, you will want to start with an angle. 80 degrees or so should be just fine, so something a bit more obtuse than "/ / / / / /" Go on and draw yourself a row of evenly angled lines. For the first little bit, you can make yourself little grids of these strokes to space your letters and follow the tilt. Rotating your paper a bit to the left also helps (my paper is nearly sideways when i write). Also: TRY NOT TO TAKE THE PEN OFF THE PAGE! That single rule really helps it look more natural. Now, what you want to do is start with the capital characters.
A good rule of thumb when trying to match the angle, is to use any vertical lines (so the centre) to match it. You'll also find horizontal lines go upwards if you write quickly, so don't be worried.
Some letters tend to actually change in makeup based on their tilt. The characters generally become slimmer and the tails longer and more wispy, and they turn upwards.
For the lowercase, I'm going to post it all handwritten, so let me know if you have problems with my (sometimes atrocious) handwriting:
(I forgot to mention that you can also write a __ under the ш to reduce ambiguity, but it's not required)
Basically, the most important thing to remember is that letters will change and some will look a bit fudged, but it's part of the stylising of your own script.
(could always be worse, you could end up with chicken scratch like this)
The most important thing to remember is to practice, practice, practice! And if you get some random non-cursive in there, that's normal too. You'll soon be writing random mumbo jumbo druid runes and not know what you've written (and it will feel good).
If you've got any questions or anything you want me to cover about Cyrillic or its pronunciation (damn you, linguistics!), feel free to bung me a comment.
EDIT: now with more