Every time you record an acoustic instrument, you are faced with a set of variables. First of all, each individual instrument will have its own sound. The performer also has her particular playing style; two different artists can make the same instrument sound completely different. Furthermore, the room--such as a studio, church or concert hall--you are recording in will have its own challenging acoustics.
Your choice of microphone and its placement will depend on the above variables. As time goes on, you may prefer certain microphones, but always make sure you bring several with you, and, if time allows, experiment with them before you start recording.
Recording engineers differ as to what style and brand microphone they use and how they place them. There is really no "wrong" microphone to use when recording piano. Producer and engineer Dan Connor recommends a matched pair of directional mics; one at the low end of the piano; one at the high end. However, Mike Senior writes in "Sound on Sound" magazine he prefers an omidirectional mic. And according to "Mix Magazine," Ed Cherney, who has recorded Billy Joel and Elton John, will use two directionals in an X-Y pattern over the hammer of the C above middle C. He'll then place an additional mic at the low end of the soundboard.
The style of music you're recording will also affect how you place your microphones. For pop or jazz, a brighter, more percussive sound is preferable. Try close mic'ing, in this case, placing the microphones a foot or so away from the strings. Keep in mind that micing too close may pick up the mechanical action of the piano.
For classical music, a more ambient sound that captures the feel of the room is preferable. Place the microphones further away from the piano to capture that room sound. However, as this adds reverb to your sound, you will have less flexibility in the mixdown. No matter what musical style you're recording, close micing is preferable in a poor acoustic situation.
If the performer is singing and playing at the same time, you need to separate the vocals from the piano tracks. Jay Newland, who has recorded Norah Jones, told "Mix Magazine" that he separated Norah's vocals from the piano by just putting a piece of foam rubber on the music stand and piano blankets on the sides. For a group recording, place the mics inside the piano to get more separation and control over the sound.
The lid of a grand piano, besides changing the volume of the sound, also points it out toward the audience. Mike Senior suggests placing mics underneath the line of the opened lid for good high-frequency recording. Removing the lid entirely is fine if you are close micing, but Senior doesn't recommend it for classical recording; it will change the overall sound of the piano.