Determining when to mulligan is a very important skill in magic. But how can I hone my abilities in that skill? Obviously if I had no/all land I'd probably want to mulligan as the opportunity to actually cast some of my spells are worth the 1 card penalty. But what about when I only have 1 land or 1 spell? Are there any good rules of thumb to help me make this decision? How can knowing my deck make these choices easier? What other elements of my hand should I look at when making this decision?
Unfortunately, knowing your deck inside out and comparing your draw against it are the only universal rules that can apply, so it's a highly individual decision that depends on those two factors. If you already have a good idea of the opponent's deck, that also helps, of course. Essentially, experience is where it's at.
You can ask yourself a few questions.
Note that if you often get into a situation where you must consider mulligan (assuming careful shuffling before each game), then there might be an issue with your deck. Adjust the number of mana sources and/or the mana curve of your spells.
Shuffle thoroughly to ensure that the deck is randomised. Shuffling helps to ensure that if clumps of land or spells do occur, they do so naturally. Pile shuffling alone is not an acceptable means of shuffling. Mash shuffling and riffle shuffling are acceptable.
Mulligans are decided upon in APNAP order. They are acted upon in parallel. First, the active player chooses whether or not to mulligan. Then, the non-active player does the same. After both players have made a decision, mulligans are taken in parallel. This repeats until one player chooses not to mulligan. Then, the remaining player takes his or her remaining mulligans alone.
Some important piece of information is born from this process. When the non-active player chooses to mulligan, the active player should have already stated his or her choice. This extra information may influence the non active player's decision to mulligan. For example, a hand full of one for one spells may be acceptable to keep against a player who is taking a mulligan, because one for one trades generally favor the player with more cards.
A solid rule of thumb people throw around is to keep the hand if it gives you a plan of action. Do you have something decent to do in the first few turns even if you don't draw what you need?
Beyond this, a good mulligan decision depends on the particular characteristics of your deck. You have to understand what cards you're likely to draw, what your mana curve is, how well it performs with a smaller starting hand, how important tempo is in the first few turns, how important it is to make all your land drops... etc.
I regularly play a couple of different (casual) decks. One is a green ramp deck, where I am often happy to keep a hand with just two lands or one land and some mana acceleration because I'm very likely to draw more mana and have very few spells that cost more than 4. On the other hand, I might mulligan an otherwise healthy hand with 3 or 4 lands but no acceleration, because getting to 3 mana on turn 2 can give me a large tempo advantage. This is very different from my other blue/red deck which has less acceleration and a steeper curve: making at least the first three land drops is critical, so I'd generally mulligan a hand with less than three lands, but worry less about tempo in the first few turns.
Apart from your own deck, you should consider everything you know about your opponent's deck. If you're facing an aggro deck with lots of burn and cheap, fast creatures, an early blocker they can't deal with could be worth more than any other spell in your deck—and you may not have many turns to draw it if it's not in your hand. But against a control deck, that same efficient blocker could be almost worthless, effectively a dead card in your hand.
Dead cards are something you should definitely pay attention to: inevitably, against certain other decks, some of the cards you're playing will be worth a lot less than others. But if you have six good cards and one dead card, mulliganing won't gain you anything, since you'd be going down to six cards anyhow! I'm often tempted to mulligan if I draw two of the same legend or planeswalker, but that's often not the best choice. (And, of course, redundant cards like this can still be useful to mitigate disruption like targeted discard, counterspells and removal.)
Apart from all this general theoretical analysis, you should also just play your deck a whole bunch of times to develop an intuition for which hands are good when. This will also help you identify weird edge case like hands which are perfect... as long as the opponent doesn't have a turn-one lightning bolt or thoughtseize. (My green deck is particularly vulnerable to this when I keep hands with a single land and a mana dork.)
In the old days, mulliganing was easy. No lands? All lands? Draw a new hand.
And then along came the Paris Mulligan - now a mulligan is always an option, not just a get-out-clause for an absolutely unplayable hand. And while you will certainly still always mulligan an all-lander or a no-lander (unless your deck is EXTREMELY unusual), there are many more borderline cases where you will have to make some tricky choices.
Hackworth has already provided a useful list of things you should be thinking about while deciding whether to take a mulligan, and I've upvoted that, but here's a couple of things that I feel compelled to add:
Beginning players, and I remember well that I used to be like this, have the mentality of "this will be a great hand... if the first card I draw is a land. What the hell, I'll keep it!" Thinking like that probably gives you a 60% chance of losing even before you play a single card, not good odds! Being a card down is bad news, but it's much less bad than running the risk of not being able to play your deck at all. Think through all the probabilities, and then, if necessary, mulligan away. I think it's fair to say that having a good sense when to mulligan is one of THE major factors that distinguishes the pro players in the game...