I know that this sounds simple, but it's really much harder than I thought it would be. Right now I have short peyos that I tuck behind my ear but I really want to grow some long curly peyos like some Chassidim (not Lubavitch) do. Does anyone have any answers? Any advice?
If you give them time and don't allow them to be cut, they'll be long. (How much time is necessary depends on how long you want them to be, of course. A tip of my hat to Isaac Moses for this link to Wikipedia, which says human hair typically grows six inches (fifteen cm) in a year.)
If you have a medical condition that prevents your hair from growing, consult a medical professional. If someone is cutting them without your knowledge while you're asleep, sleep alone and lock your bedroom door.
You mention curly, though. Some people's hair is curly; others have to curl it. I recommend curlers (also called hair rollers), not because I have personal experience with them, but because they have been quite popular for curling hair for many years so I assume they must work pretty well. However, please consult your rabbi before using curlers: I don't know whether they are considered "women's garb" of the sort forbidden to men.
My hair is curly and i could grow peyoth all day long if i wanted to. But for those who can't, then there a few things one can do, but none of them sound especially pious or manly.
Once your peyoth get long enough you can try the following:
Buy a curling iron and get curlers.
When you get out of the shower, wrap/wind your peyoth around your finger then pin the windings to your head until your hair dries. When your hair dries you can unpin it and it should be curlier and possibly have the look you're aiming for.
Do the following mentioned above but add in certain curling chemicals from a hair salon. i've even heard a chassid tell me that he mixes sugar with water and puts it in his peyoth and the sugar water helps his hair curl.
I found this interesting regarding the curling of the payos.
A man shouldn't use curlers. Rather, if you want to curl your payos, use your finger. For the proper method to do this, we asked a Rabbi with long payos. He wrote us the following: "Wet the payos and comb them out horizontally across the forehead. Then, take the forefinger of the hand of whichever side you are doing, stick the forefinger from above in between the hair and the forehead, and push it towards the roots of the payos as far as you can comfortably get it. Then, with the other hand, curl the rest of the hair around your forefinger in the down direction, trying to keep it as in order as possible. Hold it that way just for a moment, and then gently try to remove the forefinger without messing up the curl. Once it's out, don't touch the curl; let it dry that way. Do this once each weekday morning, and maybe once again later in the day if you want (optional), and hopefully it will start curling naturally by itself. "If you ever cut your payos one time too short, then about five months later you will have a growth of hair coming from the top that will not go together with the rest of the payos, at least for a couple of years. Many have this problem. I don't have a clear solution for it." Ohr Somayach - Ask the Rabbi
I have no experience with peyos (ahem), but to the other answers I'll add something learned from experience with growing hair long: conditioner.
As another answer notes, on average (head) hair grows about six inches a year. If yours just doesn't seem to be growing, it's possible that the hairs are breaking off prematurely, or splitting (split ends). I addressed this problem by using conditioner after shampooing. Conditioner isn't a hair gel that gunks up your hair all day; you rinse it out after a couple minutes.
Castor oil is great for hair: after washing your hair, put some on and your hair will grow better and stronger. Note: castor oil can get hard(er) in lower temperatures so you may need to dilute it with somehting to be able to work with it - almond oil is best.
Curling: this is NOT a priority. Payot are NOT fashion and how they look is secondary to having them - and letting them fulfil their purpose.
Having them is pretty clear where it comes from. Their function, however, is less discussed: the kabbalah tradition says [who I don't remember, sorry] that they are supposed to link your cranial hair that is connected to and expresses your intellect to your facial hair that is connected to and expresses your emotional side. This will balance your personality.
Also, hair generaly - and for men who otherwise have short hair and the only longer hairs are the payot - the payot too are, again according to the kabbalah, connected to power (remember Samson) and intuition.
So, payot should fall as naturally as possible and only minimally groomed to look orderly and not too dishelved, and touch your beard.
You can play with them - curling them around your index finger and tugging ever so slightly - but do not exaggerate: their function is to connect your head with your jaw.
Now, each Jewish community has their own tradition as to how payot should be worn - just like the kippah - and you most likely will follow that or otherwise people will start thinking you have switched to a different rabbi - thing which, in hasidic communities, can get you disinherited.
Then, do go to a Jewish barber: they most likely have lots of experience and by looking at the shape of your head, jaw and hair could advise on a particular cut that will suit you.
I have long payes, and they are curly. The main secret is to not trim them, to get long. I do curl mine, however I have naturally curly hair, it's just a matter of keeping them neat and nice looking rather than exploded and crazy. There is an awkward length where they're not really long but not short either, I don't know how long you're going for but that's something to consider. When wet, mine reach the upper part of my chest.
For curling, just get them wet and twist around your finger until tightly twisted, and pin them under your glasses for a minute. Then just twist them with your hand, pushing up then pulling down. Not sure if that's a good description. Having them curled is also important, I read somewhere that the Arizal brings down that Chochma is brought down through the curls of the payes. I can't remember the exact source though.