From the title, we can tell that the speaker is addressing this poem to a group of virgins. He's telling them that they should gather their "rosebuds" while they can, because time is quickly passing. He drives home this point with some images from nature, including flowers dying and the sun setting. He thinks that one's youth is the best time in life, and the years after that aren't so great. The speaker finishes off the poem by encouraging these young virgins to make good use of their time by getting married, before they're past their prime and lose the chance.
The poem opens with the speaker telling the virgins to gather their ("ye") rosebuds while they still can ("while ye may"). "Old Time," after all, is passing quickly ("a-flying"). The "a" in "a-flying" doesn't really mean anything; it's just an older way of pronouncing a verb. "Ye" is an old word for "your" and "you." It's not clear if the speaker is referring to actual rosebuds, or if they are a metaphor for something else. We'll have to wait and see.
The speaker elaborates on the advice of the first two lines, telling the virgins that "this flower" will die soon – although he means that everything eventually dies. Flowers don't literally smile, so the phrase likely means something like "blooms." In Renaissance usage, "die" frequently meant "have an ******." That meaning may or may not be at work here.
The speaker continues with another example of the passage of time. The higher the sun gets (the further west it moves), the "sooner" will its journey be over, because it's "nearer" to "setting."The speaker calls the sun a "glorious lamp" because it gives off light."A-getting" is just an older or poetic way of saying "getting."The sun isn't actually running a race; "race" can mean "journey, voyage, path."The progress of the sun through the sky, which is how we measure a day, recalls the first stanza's discussion of "today" and "tomorrow."
The speaker divides life into several periods and says that the "first" (i.e., young adulthood) is the best because "youth and blood" are "warmer.""Age" just means "period of time" here."Youth and blood" probably aren't literally warmer, but we often think of dead people as cold, so perhaps the speaker means something like "farther from death." Alternatively, "warmer" might even mean something like "more vigorous and healthy."
Youth is the "best" time of life, so the speaker says. Once it's gone, the "worst/ Times" follow. "Spent" means "used up" or "gone." "Worst/ Times" refers to the period after youth is "spent," so it most likely means old age or something to that effect.
Since youth is fleeting, old age sucks, and death is always right around the corner, the speaker urges the virgins to make use of what they have ("use your time") while they still can. In other words, don't be "coy," meaning shy, reserved, or inactive. The speaker doesn't just encourage the virgins to "use" their time, but to "go marry" (as in, to go get married!) while they still can.
The virgins should get married, the speaker suggests, because once they lose their "prime" (i.e., their youth and beauty) they might not get another chance. "Tarry" means "delay" or "prolong," and here the speaker wants to imply that if the virgins don't get married while they can, they might put it off (marriage) forever.