Your Handy Guide to Diagnosing Meter and Rhythm:  This is a technique that has proved effective for me in scanning verse; you may have developed your own, or may develop your own in the future,  but in the meantime, this may help you to approach the sticky job of determining meter and rhythm.
Note that the meter of a poem and its rhythm are not the same thing.  Meter refers to a pre-existing, "ideal" rhythm; virtually every poem you'll read will deviate from that rhythm at points.  Your job is then to figure out how the poem's meaning is served by that deviation.

1. First, read the poem once to determine if traditional meter and rhythm should even be considered.  If you’re dealing with free verse (as opposed to blank verse, remember), skip ahead to number 6.
 
2. If you sense the poet is working with a regular metrical pattern, check immediately to see if the poem obviously conforms to one of the traditional verse forms—it may be a sonnet or villanelle, for instance, or written in ballad stanzas or heroic couplets.  Of course, determining the meter is part of identifying these traditional forms, but you’re probably safe in predicting that a fourteen line poem with 10 syllables in each line and a regular pattern of rhyme will turn out to be a sonnet.
 
3. If the poem does seem to scan regularly, but isn’t any traditional verse form you recognize, read several lines, even several stanzas again to determine if it’s primarily a two-beat or a three-beat rhythm.  That is to say, does the meter have the tick-tock quality of a metronome, or more of a galloping quality when you exaggerate the meter somewhat?  (Note also that three-beat meters generally “feel” faster because of the succession of unstressed syllables.)  It helps to pause at the end of lines, even run-on lines; it also helps to ignore the actual sense of the words and exaggerate the sing-song quality of the rhythm.  It might even help to tap out the rhythm with your hand or your foot--this is very much like determining rhythm in music, after all.  Most poems in English are of the two-beat variety, but you should always read carefully, rather than jump to conclusions.
 
4. Once you’ve determined if the meter is a two-beat or a three-beat meter,  decide whether a two-beat meter is primarily iambic or trochaic, or whether a three-beat rhythm is primarily anapestic or dactylic—that is, ask yourself where the (relative) stresses fall (remembering, again, that very few poems are completely consistent throughout).
 
It may help to keep a line of each in your head for comparison:

A few more hints: feet that end on a stressed syllable are sometimes referred to as “rising feet,” and, not surprisingly, feet that end on an unstressed syllable, “falling feet.”  You can sense the distinction in the lines above (I think, anyway).  And the three-beat meters are often used in light verse--the anapest in the limerick and the dactyl in the double dactyl, for instance--though they’re also employed for serious verse.
 
5. Once you’ve determined the metrical pattern, the real fun begins: recognizing where and how the poem’s own rhythm deviates from the underlying meter, and to what effect.  THIS IS NOT AN EXACT SCIENCE—even if some instructors think it is.  There are many disagreements over how to scan specific lines.  Try exaggerating the presumed meter as you read, noting the places where it seems to distort the natural inflections of spoken English.  “On First Looking into Chapman’s Homer” is a sonnet, which means it should be iambic pentameter, but try exaggerating the meter  as you read the first line: That sounds okay, except that stressing “have” more than “much” sounds strained; in ordinary speech, you’d emphasize “much” rather than the auxiliary verb.  So you’d probably want to scan
 the line (The power of the meter will probably contribute a slight stress to the normally unstressed preposition “in,” but probably not as much as more important words, like “realms” or “gold.”)  The next question is, what does this deviation from the prescribed meter do to our experience of the poem?  And for that, you have to consider the line in the larger context of the poem as a whole.
 
Words of single syllables are often hardest to figure out, relative to the other words in the line, but words of two or more syllables have natural inflections that should take precedence over the underlying meter.  So “silent,” in the final line of the Keats's sonnet, doesn’t become siLENT; you ascribe to it its proper inflection, SIlent, but against the background expectation of the iambic foot.
 
6. If you’ve identifed the poem as an example of free verse, scanning is an even thornier issue; some critics contend that all good free verse plays with conventional meter, even if only to disrupt it.  This is certainly true of some free verse (many of T.S. Eliot’s works, for instance).  In other cases, however, free verse really is just that, and you may find so many exceptions and deviations that relating the poem to any standard meter will seem extremely dubious.  Do listen for the echoes of standard meters as you read a free verse poem, but recognize that you may be looking for something that’s not there; you may be better off considering line length, or the lines’ relation to conversational language, or some other rhythmic principle.  Sometimes patterns of rhyme, including near-rhymes like assonance and consonance, will influence the rhythm of a poem in free verse, as in Swenson's "Riding the 'A,'" where internal rhymes and near-rhymes contribute to the speed of particular lines.  There are also rhythmic patterns that may seem initially like free verse but which do follow a prescribed pattern; the easiest of these to spot is syllabic verse, where the poet has counted syllables with no regard to stress.  For instance, every line of the poem may contain, say, 12 syllables, or the number of syllables in each line may be consistent from stanza to stanza (such as a poem of three-line stanzas where the first line of each stanza has five syllables, the second line seven, and the third line nine).
 
A related approach is sprung rhythm, where only the stressed syllables count; in sprung rhythm,  you might have a seven-syllable line and a ten-syllable line, but the lines will be consistent in having five stressed syllables.  This is a very old approach to poetic rhythm in English, going all the way back to Anglo-Saxon verse, but it’s notoriously difficult to scan; in fact, Gerard Manley Hopkins, who coined the phrase “sprung rhythm,” often used accent marks to indicate where the stresses should fall.