This answer is about whether the Milky Way is gravitationally bound to other galaxies or clusters around it. I did answer a question that delved into whether the Milky Way Galaxy was orbitally bound to Sagittarius A* here. It is actually an answer to whether a supermassive black hole was at the center of the Milky Way, or any galaxy for that matter, and what would happen if it disappeared somehow. It shows a lot of gravitational math.
I did the calculations between the given mass of the Milky Way, and the given mass of the Andromeda galaxy. The 2 galaxies are pulling at each other with 114 quadrillion teranewtons. That sounds like a big number but that translates to the Andromeda galaxy accelerating us toward it at a rate of 5.76 X 10^-14 meters per second squared. That is 1/3000th the distance from the nucleus of a hydrogen atom to its orbiting electron (Bohr Radius).
By Isaac Roberts (d. 1904) - A Selection of Photographs of Stars, Star-clusters and Nebulae, Volume II, The Universal Press, London, 1899., Public Domain, File:Pic iroberts1.jpg. So cool seeing this picture from 1899!
Over a year that means that the path of the Milky Way will be deflected 28 meters, and increase its velocity toward the Andromeda galaxy by 0.0018 millimeters per second. What that means is that the 2 galaxies are ballistic at 300 kilometers difference in velocity and not orbital in trajectory. They would pass by each other and keep getting farther apart forever if they were not going to collide somewhere around 4.5 billion years from now.
One thing we do know is that there are several dwarf galaxies circling the Milky Way that are in orbit (Satellite galaxies). There are 59 dwarf galaxies that are close enough to be in orbit of the Milky Way, but some are moving too fast and are ballistic, and others could be in orbit of other dwarf galaxies.
Exploring the Local Volume in Simulations
The local group is a series of 54 galaxies including the Milky Way and the dwarf galaxies that orbit it that are in close proximity. Most are gravitationally bound to the Milky Way, Andromeda, or Triangulum, and Triangulum may be bound to Andromeda, but the Milky Way is not in orbit of another galaxy.
The Virgo Supercluster is a cluster of over 100 galactic clusters.
Virgo Supercluster - Wikipedia
These all could be gravitationally bound, but the movement of galaxies that far away is extremely hard to detect beyond the red shift which only tells the observer whether it is approaching or receding from us. It does not show lateral velocity. We may be able to tell some of these velocities in a hundred years when we have detailed photos to compare movement.
The Virgo Supercluster could be part of the much larger Laniakea Supercluster which is theorized to be circling the Great Attractor. If it does exist or if it is the center of gravity of the Laniakea Supercluster, then we are in orbit of this Great Attractor. That orbital period though can only be theorized, and it would be over a billion years if not tens of billions of years.
By IPAC/Caltech, by Thomas Jarrett - "Large Scale Structure in the Local Universe: The 2MASS Galaxy Catalog", Jarrett, T.H. 2004, PASA, 21, 396, Public Domain, File:2MASS LSS chart-NEW NASA.jpg
The blue arrow to the lower right points to the location of the Great Attractor.
Updated 11/8/18
Another option rather than orbiting the Great Attractor, or the barycenter of the Laniakea Supercluster is that the Milky Way is affected more by the local galaxies around it, and as such follows a random path as other galaxies approach and move away with the Great Attractor adding its own influence if it exists. We just don’t know or have any proof of the Great Attractor’s existence or what it might be otherwise I would add its influence in gravity to this answer.