I second (third?) helmstead and kstaven -it sounds suspiciously like mastitis (clotted old blood would look brown) or a severely congested teat and that lady definitely dumped a problem on you (shame on her!). A vet is likely your best option at this point, GOOD LUCK!
From Storey's Guide to Raising Dairy Goats:
Mastitis
Mastitis is inflammation of the mammary gland, usually caused by an infection. Symptoms are a hot, hard, tender udder; milk may be stringy or bloody. (Routine use of a strip cup before milking will alert you to abnormal milk; see chapter 13 for details.) Mastitis may be subclinical, acute, or chronic. It's usually a relatively minor problem, but some forms, such as gangrenous, can be deadly.
Not all udder problems indicate Mastitis. Hard udders (typically just after kidding) that test negative for mastitis are referred to as congested, and usually disappear. Congested udders are best cleared up by letting the kids nurse, and massaging the teats and udder for 3 to 4 days after parturition. In mastitis, the alveola or milk ducts are actually destroyed. Since it's necessary to identify the bacteria involved, the services of a vet are required.
Mastitis can be caused by injury to the udder, poor milking practices, or by transference by the milker from one animal to another. Hand milkers should wash their hands between each animal, and udders and teats should be washed and dried before milking. Teat dips have proved of great value in controlling the disease among cattle, although the solutions must be diluted for goats.
Somatic cell counts, used to detect mastitis in cows, are not considered reliable with goats. (See page 254)
Home tests for mastitis are available from veterinary supply houses. The best-known is the California mastitis test, or CMT. However, Smith and Sherman in Goat Medicine say, "The CMT is more useful in ruling out than for diagnosing mastitis in goats. In a well-managed herd, the predictive value of a positive test is unacceptably low."
Prevention
As you might expect, an ounce of prevention is worth a pound of cure. Dry, clean bedding, especially at freshening, is important. Wash and dry udders with individual towels for each animal. Use a spray bottle for the sanitizer solution, rather than a communal bucket. Milk with clean, dry hands. Milk gently, in peaceful surroundings. Avoid vigorous stripping.
More:
The first squirt from each teat should be directed into a strip cup, a cup with a sieve or a black plate for a cover. That first stream is high in bacteria that have collected in the teat orifice and it shouldn't go into your pail. In addition, the use of a strip cup will enable you to see any abnormalities in the milk, such as lumps, clots, or stringiness. This is an indication of mastitis, which demands your attention (see chapter 8.) Never use the milk from any animal that's not in perfect health.
