What kind of hydrangea is it? Big leaf? Panicle? Oakleaf? Smooth? This would determine treatment.

You have what is called a rooted cutting. No matter what kind of hydrangea, rooted cuttings do not often grow large enough to bloom for 1-3 years.

In what region do you live? Anywhere below USDA zone 7 should, rooted cuttings of hydrangea macrophylla (big leaf) should be pot-grown for the first couple of years, to reduce risk of not surviving winter. The pot size should be 2-gallon or larger. To properly "start" the cutting, you need to keep it in a moderately shaded area ("somewhat dark", NO direct sunlight) and the soil kept continually moist. Shade + regular rainfall should accomplish this. Over winter, the pot should either be kept above ground in temps ranging from 32-45 (garage/basement) or below ground IN the pot with a small pile of dead leaves over it. Remove leaves in April, or set pot back outside after danger of frost/freeze has passed.

If not a bigleaf, the other main types could probably survive outside planted normally, like any perennial.