Red veins or red streaks are usually associated with ammonia/nitrite spikes in the water. Ammonia  and nitrite levels can rise suddenly in a newly setup aquarium or one that is overcrowded (too small or as a result of overfeeding). I suggest you do a large water change around 40-50 percent now to bring down these levels down and continue small water changes for next few days until the levels are back to normal. Go buy and aquarium test kit and test for ammonia, nitrite and nitrate levels. Be sure to use a water conditioner/dechlorinator  with any new water you add to the aquarium and make sure the temperature of any water is about the same as the water currently in the aquarium.

The white spots are probably ich, which can be cured by raising the water temperature to 83F or so and add aquarium salt at 1tsp  per gallon or if you prefer medications - Jungle labs or Mardel Ich meds.

What is the size of your aquarium? If it's relatively small, you will have a lot of problems with ammonia , because goldfish excrete a huge amount of wastes.

You seem to have several problems going on here.  The red veins could also be septicemia (an internal bacterial infection).  First check for ammonia levels with a good  test kit. If it is present , then you need to determine what is the cause and fix it.  If there is no ammonia present , then it's most likely a bacterial infection which you can treat with Maracyn TC or Maracyn 2.