Openings that make sense

Five principles

You do not need to memorise long lines to play the opening well. Follow these and you will reach a healthy middlegame in almost every game.

Fight for the centre. The squares e4, d4, e5 and d5 are the most valuable on the board. Occupy them with pawns or control them with pieces.
Develop your pieces. Bring out knights and bishops before moving the same piece twice. Knights usually before bishops.
Castle early. Get your king out of the centre and connect your rooks, normally within the first ten moves.
Do not bring the queen out too early. She will be chased around by smaller pieces while they develop with tempo.
Do not grab pawns at the cost of development, especially wing pawns. Time matters more than a pawn in the opening.

A model openingAfter 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.c3 Nf6 5.d3 d6 6.O-O O-O, both sides have followed every principle: central pawns, knights and bishops out, kings safe.
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Traps to know

Scholar's Mate

White brings out the queen and bishop early to attack f7, the weakest square in Black's camp because only the king defends it.

Scholar's MateAfter 1.e4 e5 2.Bc4 Nc6 3.Qh5 Nf6?? White plays Qxf7#. The fix is simple: 3...g6 or 3...Qe7 defends, and the white queen will soon be chased away.
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Legal's Mate

Legal's MateBlack pins the knight with ...Bg4 and then greedily grabs the queen. 1.e4 e5 2.Nf3 d6 3.Bc4 Bg4 4.Nc3 g6? 5.Nxe5! Bxd1?? 6.Bxf7+ Ke7 7.Nd5#. Pins are not always as strong as they look.
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Fool's Mate

The shortest possible game: 1.f3 e5 2.g4?? Qh4#. Moving the f and g pawns early opens a highway to your king.

A repertoire for White

1.e4 and the Italian Game

Open with 1.e4. Against 1...e5 play 2.Nf3 Nc6 3.Bc4, the Italian Game. The plan is c3, d3 or d4, castle, then Re1 and Nbd2-f1-g3. It is easy to learn and grandmasters still play it.

Italian GameThe bishop on c4 aims at f7. White will follow with c3, d3, O-O and a slow build-up.
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Against the Sicilian (1.e4 c5)

The open Sicilian is a lot of theory. A practical choice is the Alapin, 2.c3, planning d4 to build a big centre. Against 1...e6 (French) or 1...c6 (Caro-Kann), the Advance Variation, 3.e5, gives you space and a clear plan.

Prefer 1.d4? Try the London System

With 1.d4, 2.Bf4, e3, Nf3, c3, Bd3, Nbd2 and castle, you can play almost the same set-up against everything. It is solid and saves study time.

London SystemA typical London set-up. The dark-squared bishop comes out before e3 so it is not shut in.
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A repertoire for Black

Against 1.e4: 1...e5

Meet 1.e4 with 1...e5. You learn the classical ideas that every strong player knows, and the positions are fair and open. Against the Italian, play ...Bc5, ...Nf6, ...d6 and castle. Against the Ruy Lopez (3.Bb5), 3...a6 4.Ba4 Nf6 5.O-O Be7 is the time-tested main line.

Ruy LopezThe Spanish Game, 3.Bb5. White puts pressure on the knight that defends e5. 3...a6 asks the bishop a question straight away.
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If you want something more solid, the Caro-Kann, 1...c6 followed by ...d5, gives you a strong pawn chain and a good light-squared bishop.

Against 1.d4: the Queen's Gambit Declined

Answer 1.d4 d5 2.c4 with 2...e6. Black holds the centre, develops with ...Nf6, ...Be7 and ...O-O, and frees the position later with ...c5 or ...e5. It has been a world championship regular for over a century.

Queen's Gambit DeclinedBlack declines the c4 pawn and keeps a firm grip on d5.
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How to study openings

Learn the ideas and plans before the moves. After every game, look up the first moves where you or your opponent left the book.
Use the Opening card in the analyser: it names the opening and shows the main book moves from any position.
Below about 1500, opening knowledge decides very few games. Tactics decide most of them. Spend most of your time on tactics.