For long time periods, hydrangeas do not do well indoors. Consider keeping potted ones indoors while they are dormant but bring them out at an appropriate time for your location... two weeks after your average date of last frost in Spring. Any time that the plant is dormant is also a good time to repot but I prefer to do it closer to Spring in case repotting or cutting of roots causes the plant to wake up unexpectedly.

The bugs could be spider mites. Miticides (such as Spectracide Permethrin Insect Control, Bonide Kelthane, Schultz Houseplants and Gardens Insect Spray and others) can be used but follow the instructions on the label; they are best used alternating with other types of pesticides, such as soaps and oils (like Safer Insecticidal Soap & Sunspray Ultra-fine Spray Oil). This prevents the mites from developing tolerance to the miticides. I prefer to use the soaps, followed by the oils, before using the big guns (miticides). You could also make a habit to apply these products about 2-4 weeks before bringing any pots into the house (regardless of whether there are visual confirmation of pests or not).

Hydrangeas are NOT houseplants :-)) While they are frequently sold offseason in full bloom at grocery stores and florists, these are plants that have been grown in greenhouses under highly controlled conditions and generally have a pretty short indoor life under normal household situations, as Luis has indicated. These are typically considered as throw away plants once they have finished blooming unless you can transition them safely to the landscape.

Ideally, in most places in the northern hemisphere, hydrangeas should be dormant now. A plant in leaf and/or bloom indoors is out of the normal growth cycle and will suffer various sources of stress as a result. Indoor condition of most homes are just not suited to keep the plant healthy and insect infestations, discolored foliage and the dropping of leaves will ensue. Webbing indicates the presence of spider mites and lack of humidity (one of the most common problems of attempting to overwinter landscape plants indoors). Water will remove the mites (give the plant a good shower in the kitchen sink or bath) and you may want to mist regularly after to keep up humidity levels......spider mites like dry conditions and abhor moisture and high humidity.

I agree with Lius to hold off on repotting until closer to spring and the plant's normal growth cycle. Since you are in a much colder zone than most bigleaf hydrangeas are happy in, you do probably want to continue this as a container plant, however it needs to go through as much a normal growth cycle as possible for long term viability. Put outdoors when the weather moderates and then overwinter in a protected location, but not inside the house. It will need a normal period of dormancy with temperatures right around freezing, but not much lower. A crawl space, unheated basement or garage (if it stays right around freezing) will work. Light is not an issue during the dormant period (no photosynthesis taking place) but it will require modest moisture, just enough to keep soil barely damp.

Thanks for your help everyone. It could not be an outdoor plant where I live (especially the kind of hydrangea I received), so I thought I would give it a try anyway. Can't fault me for trying, right?