“methinks I have astronomy”
Here 'astronomy' means astrology — the practice of reading the positions of the stars and planets to predict future events. In Elizabethan English the two words were largely interchangeable; the modern distinction between observational astronomy and predictive astrology had not yet fixed itself. The speaker claims to possess this art but immediately qualifies it.
“By oft predict that I in heaven find”
'Predict' here is used as a noun, not a verb — it means a prognostication or omen. 'Oft predict' means a frequently observed celestial sign. The speaker says he cannot advise princes whether things will go well for them based on the omens he finds in the sky.
“to store thou wouldst convert”
'Store' means offspring or reproductive increase — to 'convert to store' is to turn from self-absorption toward procreation, investing one's gifts in the next generation rather than keeping them only for oneself. This is the core of the procreation argument that runs through the first seventeen sonnets.
historical The procreation argument (Sonnets 1-17)“truth's and beauty's doom and date”
'Doom' means judgment or the appointed end — not primarily disaster, but a final, irreversible verdict. 'Date' means the appointed time or term of existence (as in 'out-of-date': the time has expired). Together 'doom and date' form an alliterative phrase for the absolute ending of something. If the youth dies childless, his death will be the death of truth and beauty themselves.