Avoid reading the label on a piece of art when you first see it. Try to understand the artwork before letting something else affect your opinions.
Get a first impression by assessing if the artwork evokes any emotions, such as sadness, happiness, fear or confusion (this is normal).
Look at the lines. Assess whether they are smooth and soft like Renoir paintings or dark, hard lines. Look at how the artist used lines; for example, Van Gogh used lines to show wind currents in his starry night, while Munch uses lines in his Scream to show sound waves.
Note the colors and tones and how those create the mood of the artwork.
Look for possible symbols, especially in religious paintings. A dove often represents the Holy Spirit while a snake symbolizes evil. The more you analyze art, the more you'll recognize and understand these symbols.
Notice the composition and perspective of the artwork, including whether it is busy or symmetrical and the angle. For instance, in Tintoretto's Last Supper, the composition and perspective is on an angle from above, which is different from most paintings of the Last Supper, which are usually symmetrical and centered.
Notice the lighting and shading. Often artists use light for a dramatic effect, like in Carravaggio's painting The Conversion of St. Paul. Light can tell also you the time of day the painting takes place and if the light is from an artificial (lantern) or natural source (lightning).
Use all the aspects you've gathered so far to understand the theme. The theme is the message of the painting.
Now look at the label. The label should at least tell you the name of the artwork, the artist and the date. Sometimes a label also indicates what the art piece about. For instance, the famous painting Whistler's Mother is actually called Arrangement in Grey and Black: The Artist's Mother. This shows that the artist was concentrating more on the color arrangement than his love for his mother.