Most people select cuttings that have no blooms and cut the leaves in half. You can apply a medium root stimulator and plant them on potting mix. See the link with good instructions below.

Here is a link that might be useful: Propagating hydrangeas from cuttings

Rather than taking a cutting from this plant, what would be the best way to encourage the plant to grow?

When I have large cuttings, I use some root stimulator and get rid of bloomage so the plant concentrates on developing a root system instead of blooms. I try to keep the soil moist but not wet in well draining pots that have at least two holes and a layer of rocks at the bottom. The more cuttings you have, the more chance of at least one growing ok. If the mother plant is still there, some people try to do ground layering as a form of propagation. See the link above about that.

Dave, not sure where you are located - that has a bearing on how well the plant will do on an ongoing basis.

What you have is a florist's or greenhouse grown hydrangea - one that is forced into bloom out of season and at a very young age. These can be transitioned out into the garden - easier in the warmer zones (7, 8 & above) but also possible in Z5 & 6 with winter protection.

Keep the plant in bright but indirect light indoors in a cool location. Keep the soil moist but not wet. The flowers should last quite awhile but you can remove them if they start to lose color or wilt. Just cut off below the flower head above the first set of leaves.

After all chance of frost has passed you can transition the plant out of doors. Put it in a shady, protected location (covered porch?) for a few hours a day, gradually increasing that time period to a full day and eventually, overnight. You can safely plant it outdoors at that time, following whatever practice is appropriate for bigleaf hydrangeas in your climate.

It is unlikely the plant will rebloom this season but if treated carefully and given adequate winter protection as necessary, it should bloom in season next year.

Thanks for all of the information! I live in Pittsburgh, so zone 6B from what I've gathered. I'm hopeful that the last frost has past, but I will begin to set the plant out on the covered porch now during the day. Is the size of the pot critical in any way? I'd like to avoid having to change it for a while so is there any negative to planting into a larger pot?

Move it to a larger pot as soon as you can, if you want to grow it there for a year or two. Try a plain or decorative 3-gallon container.

Feel free to enjoy the blooms until they dry, then cut down to the portion of each flowering stem just above the first set of large leaves.

You might have a reblooming variety here, so you might just get flowers again later this summer.

Wow, that blue is so beautiful!