First and foremost, there is no magic formula for getting better at chess. It takes work, a lot of work. Professional chess players study for upwards of 8 hours per day (it is their full-time job after all). Point is, you can study chess for a lifetime and still never know everything there is to know about the game because of its rich complexity. But there is no shortcut to getting good.
That being stated, there are some principles of the game that can get you pointed in the right direction so the work you put in actually means something.
1 - Learn how to Checkmate
The object of the game is checkmating your opponent's king. Many really good players often lose sight of that objective. So before you start getting wrapped up in the Sicilian dragon opening theory, or the Evans Gambit, or studying the subtle interplay of knights vs. bishops in closed positions, learn to checkmate the king. Fortunately, learning to checkmate the king is like riding a bike. Once you galvanize the patterns in your mind they are there forever. Every new player I coach in chess, we start with checkmating the king.
May I recommend this website, called "Mate in One" which is nothing but "White (or black) to move and checkmate." It's simple, but will teach you a lot about the game of chess.
2 - The Bigger Army Usually Wins
If understanding the objective of chess is to checkmate the opponents king, it's important to know the side with the larger army can usually beat his opponent into submission, then checkmate the king.
Everyone knows the queen is the most powerful piece on the board. But is a bishop stronger than a rook? Are two rooks stronger than a queen? What about a knight and a pawn for a rook, would you make that trade? What about three pawns for a bishop, would you make that trade? A general guideline to help you make these decisions are piece values. All pieces are rated in terms of how many pawns they're worth. A pawn is worth one pawn, obviously.
Pawn = 1
Knight = 3
Bishop = 3
Rook = 5
Queen = 9
King = Infinity (Don't trade the king. If you do that it's game over.)
Knowing these piece values can help guide you to making good decisions during the game. If I can give up my rook and capture my opponent's knight and bishop I should do that because the rook is worth five and a knight and bishop combined are worth six. Therefore I come out ahead. Staying on the "advantageous" side of all of your trades will lead you straight to a lot of victories.
3 - Give Yourself a Lot of Options
If the main objective of the game is capturing the king, and having a bigger army will almost always lead to checkmate, then having a lot of options will give you good chances at getting the larger army and/or capturing the king. What I mean by giving yourself a lot of options is getting your pieces into the game. A big mistake a lot of beginners make is to mobilize one or 2 pieces onto the board and then start attacking while the rest of their army is sitting back at home base doing nothing. Get your pieces (not pawns) off of their home squares. In the first ten move of the game, you should move all of your pieces from their starting squares towards the center of the board in some fashion, castle your king, and touch your rooks together (If you don't know what Castling is, look up a tutorial on YouTube. It's essential knowledge). Having all of your forces mobilized will give you a lot of different and exciting things you can do. Your force is always strongest when it's working together in concert rather than operating independently.
4 - Don't make Idle Threats
Disclaimer: I realize this is the ugliest chess board ever. I color-coded it so you could follow along with my example.
It's white to move. So what's going on here? Well material is even. That means the armies are the same size. White's rook is being attacked on d5 (blue) by a knight, so he needs to move it. White also has a checkmate threat. White is threatening to move his queen to g5 (orange) and checkmate black on g7 (red) because the queen is supported by the bishop on b2 (lime green).
A weak player, would see that threat and immediately move to Queen to g5 (orange). But that's a bad move, because black can respond with pawn to f6 (yellow) blocking the path of the bishop, and threatening to take the queen. Now the queen has to retreat and then black is going to follow up by taking the rook on d5 (blue) with his knight, winning an exchange (giving up a knight, 3, and gaining a rook, 5) and black will have a big advantage.
That's what a weak player would do.
An intermediate player would see that does not work, and would likely retreat his rook back to d1 (purple). Not a bad move, but there's better.
A good player will see there's a threat to mate by playing Queen to g5 (orange), and she'll look for a way to take advantage of it. The right play here is Rook takes knight on a5 (gray). Black will follow by bishop takes Rook on a5 (gray) and then white will now follow with Queen to g5 (orange) simultaneously threatening checkmate on g7 (red), and the undefended bishop on a5 (gray). At worst, white is going to come out on top by a pawn (he gains a knight, 3, and a bishop, 3, in exchange for a rook, 5). At best, black will overlook the mate threat and retreat his bishop and white will win by playing Queen takes pawn at g7 (red), checkmate.
I don't want to get you bogged down in analysis and theory and fancy play. I just wanted to illustrate that just because you see a threat, don't grab it. Be patient and look for something better. There's an axiom in chess, "When you see a good move, try to find a better one." Even strong players sometimes fall victim to making a good move instead of a great one.
5 - Play a lot
Sign up for chess.com. It's free, they have a smartphone app, and you can play to your little heart's content. As I alluded to in the beginning, getting the crap kicked out of you will help you learn a lot. When I first started playing, I fell victim to a lot of traps. But that helped me learn the traps and also how to avoid them and then finally how to beat them. I would urge you not to play computers. Computers don't make the same kinds of mistakes humans do. Chess engines have to force themselves to make bad moves at random intervals. Playing against people will help you learn how and why people make easy mistakes, especially when they're under pressure and will condition you to play a certain way. You'll get no such luxury from a computer.
6 - Don't Give Away Your Army
One bad mistake people make is they simply overlook getting their pieces taken. If someone moved a pawn up and threatened your knight... move your knight. This seems mundane, but especially early on in getting better at chess protect your army! If you focus on this and this alone and not even worry about checkmating the king, you will win a lot of games. The reason is because most people aren't disciplined enough to care for their army. Eventually your opponent will screw up and leave a nice juicy knight just sitting wide open on the board for your queen to take for free.
Be disciplined and alert. The person who wins the game of chess is the one who makes the next-to-last mistake.
7 - It's a Game, Don't Sweat it.
Look, you don't have to see 20 moves deep into the future to be a good player. People often think that chess grandmasters have these mystical powers of computer-like analysis. They don't. Some are better than others, obviously, but for the most part they simply have a lot of experience, know a lot about the game, and can sometimes play without even calculating moves they have so much experience. Learning the principles of the game, exercising some patience, and discipline will go a long way in aiding you to becoming a proficient chess player. And even if you don't get any better, don't worry about it. It's just a game. It's been said that knowing how to play chess is the mark of a gentleman. Knowing how to play chess well is the mark of a wasted life. It's a joke, but do keep that in mind. The game is there to be enjoyed.
8 - Get Advice from Better Players
Most players who are any good at all won't mind spending some time helping weak players. The reason is because someone helped them when they were a weak player and they remember what it's like. Don't be afraid to ask someone for help or advice who is better than you. You can even ask me for advice. I'm available on Reddit @weaksquare (yes, it's a chess term). Happy King-hunting.