“How heavy do I journey on the way”
'Heavy' carries two meanings at once: sorrowful (as in 'a heavy heart') and physically burdened or slow-moving. The same word applies both to the poet's grief and, by the next line, to the literal weight his sorrow places on the horse beneath him. OED records 'heavy' in the sense 'weighed down with grief, sorrowful' from Old English onward (OED s.v. 'heavy,' adj., sense 6a).
“ease and that repose to say, 'Thus far the miles are measured from thy friend!'”
The syntax is compressed: 'what I seek' (the destination, the hoped-for rest at the end of the journey) 'doth teach that ease and repose to say' the quoted words. In other words, the very thought of arriving — of resting — can only remind him that each moment of arrival is also another measure of distance from his friend. 'Measured from' means counted off in miles away from, not toward.
historical The Fair Youth, the Dark Lady, and the Rival Poet“The beast that bears me, tired with my woe, Plods dully on”
The horse is given a sympathetic imagination: it 'plods dully' not from its own weariness but because it has absorbed the rider's grief, and it seems by 'instinct' to know its rider would not want to arrive quickly. 'Being made from thee' means 'being separated from you' — 'made from' is an Elizabethan idiom for 'removed from,' 'parted from' (OED s.v. 'make,' v., sense 62b, 'make from': to go away from). The animal's slow pace is thus a physical embodiment of the poet's reluctance.
historical The sonnet form“The bloody spur cannot provoke him on”
'Bloody' here is literal, not an oath: the spur is one that actually draws blood from the horse's side — a severe, rowel spur used in desperation to force a reluctant animal. 'Provoke' keeps its Latin sense of 'to call forth, to goad' (Latin provocare), stronger than the modern word. The image concedes that the poet does try to urge himself forward, but the grief undermines the very instrument of urgency.