In water, $\ce{H+}$ never exists, the proton is always part of some other chemical (or ion). So, it would be a bit incorrect to say that $\ce{H+}$ bonds with water, at least in the sense that a reaction has occurred between a species best described as $\ce{H+}$ and something else (water).
I am tempted to claim that the answer by Bz is wrong, but it's reasonably close to being correct so will refrain. $\ce{H+}$ in water exists as both the $\ce{H3O+}$ ion and other $\ce{H + $n$\ H2O}$ species such as $n=2$ $\ce{H5O2+}$ and $n=3$ $\ce{H7O3+}$. These latter can be thought of as polymeric ions. (Because of charge repulsion, doubly negative species are a negligible component of liquid water.) The solvated (or aqueous) proton is generally just called the hydrogen ion, but other names are certainly possible. $\ce{H+}$ is the most common species in the Universe, but it doesn't exist in water. Since, for reactions in solution, the solvent is quite important, but beginning students are usually 'protected' from this level of detail, $\ce{H3O+}$ is a good way to express the concept how the $\ce{H+}$ (reactive) species is being 'carried' (or where it has gone to). Keep in mind that the hydrogen bonding that goes on in water is important for its physical and chemical properties. Any real reaction involving the transfer of $\ce{H+}$ in water will be similar to $\ce{XH + Y -> HY + X}$ (where these species may be charged or neutral (of course charge will balance) and definitely not involve $\ce{XH + Y -> XY- + H+}$, but the last could be $\ce{-> XY- + H3O+}$ (with the understanding that $\ce{H3O+}$ is shorthand for $n=1$ mostly with a bit of $n=2$ and less $n=3$, and less and less $n=4,5,6,7$).