Can you be more specific about what you are looking for? There was a lot going on between 1590-1610, including a war with Spain, the Essex Rebellion, the death of England's then longest reigning Monarch, the ascention of a Scottish King to an English Crown, cessation of hostilities with Spain, four Parliaments, the advent of Colonization in the New World, the continued colonization & conquest of Ireland, etc.

As far as primary sources go, there is a abundance. The ones that quickly come to mind are:

Sorlien, ed. The Diary of John Manningham of the Middle Temple, 1602-1603

Renold, ed. Letters of William Allen and Richard Barret, 1572-1598

Marcus, ed. Elizabeth I: Collected Works

The various collections of State Papers, either Domestic, Venetian, or Spanish are online for free; Privy Council Registers are also available (Though a fire in 1618 destroyed most of the Registers for James' early years). The sources available all depends on what you are looking for.