The very first line of the poem "Come live with me and be my Love," (line 1) implies the youth of both parties. Since neither of them currently has a spouse (and are thus free to marry each other) it follows that they both might be in their young adulthood. The rest of the stanza invites the maiden (or nymph) to whom the Shepherd is speaking to enjoy the "pleasures" of the natural landscape. It is assumed here, also, that it is clement weather in which to enjoy these scenes. Spring would be one season in which it would be enjoyable to roam the "hills and valleys, dale and field,/And all the craggy mountains" (3-4).
Stanza two becomes more explicit in its indication of time of year. "Melodious birds sing madrigals" (8) definitely conjures up a mental image of springtime. Birds do sing other times of year, but by far season with the most birdsong, in England, is spring.
"There I will make thee beds of roses/And a thousand fragrant posies" (9-10) continues the suggestion of springtime, when flowers are much in bloom. The rest of the the third stanza confirms this, with further references to "A cap of flowers, and a kirtle/Embroider'd all with leaves of myrtle." (11-12)
The fourth stanza contains the first reference to "lambs" (14) which now definitely places the time in spring, when lambs are born. But this stanza also contains an opposite reference to "cold", implying that it is (or will be) a cold season. This could refer to cool spring nights, but generally cold implies the season of winter. This could, however, be wishful thinking on the Shepherd's part; he could hope that his love would still be with him after the spring, summer, and fall, and into the following winter.
Stanza 5 contains another foliage reference "ivy buds" (17), which would occur in the springtime. But in the last stanza, the month of May is definitely mentioned (21), so now there is no doubt that the Shepherd is talking about a springtime wooing. Nowhere in the poem is the youth of either party directly referred to, but it can be construed that the finery the Shepherd conjures up (cap of flowers, embroidered kirtle, gown of lambswool, gold-buckled slippers, and a belt of straw, ivy, coral and amber) would most likely be for a young woman. The suggestion of youth, however, is mostly implied and to be understood through the conventions of pastoral poetry. Shepherds in pastoral poetry were always young, and their mistresses corresponded in age. The light-hearted tone of the piece also implies the speaker's youth.
A fine rebuttal to this poem has already been written by Sir Walter Raleigh (see link below "Nymph's Reply to the Shepherd", also called "Answer to Marlowe") in which the nymph (or maiden) replies to each fantastic claim by the shepherd with level-headed practicality. To consider this to be a serious and literal offer seems disingenuous; this is poetry, and to read it only literally denies much of poetry's meaning. The Shepherd, imaginary though he may be, is expressing his love as best he knows how, and is offering homage to his beloved. The maiden's feelings, however, would depend on if she believed literally that a shepherd would provide her with these rich gifts, and with a fantastically utopian and quite impossibly luxurious rural life, or if she took it rather as a poetic expression of devotion.