How fast would we have to go for the red(/blue)shifts of galaxies ahead of us to differ from those behind us with statistical significance? Here, we are looking for a statistically significant difference between the mean redshifts of galaxies ahead and those behind, with reference to a prior-specified direction. For example, suppose that we were aboard an advanced spaceship that had been accelerating in that direction at (say) 1 $g$ for some years. How soon into our journey would we notice that the galaxies ahead were bluer than the galaxies behind? Clearly this depends on our existing model, based on prior observations, for the distribution of galaxies' redshifts, and on some reasonable kind of weighting for each individual galaxy (perhaps according to the cosine of angle from our direction of travel, but not according to the galaxy's distance). However, in this analysis, we are ignoring individual data about any particular galaxy acquired before our present observation. For simplicity, let us also suppose that there is no statistically significant difference between the mean redshifts forward and back in the line of our journey before we began it. For the sake of definiteness, if it is needed, let us take a significance level of 1%.
First off, I'm not going to calculate an answer to the "significance level of 1%" part. This is more a rule of thumb kind of answer.

We're moving in a spaceship and want to see


  a statistically significant difference between the mean redshifts of galaxies ahead and those behind


In other words, the red- and blueshift due to our motion must exceed the peculiar motion of the individual galaxies. Cosmologists have been dealing with a very similar problem trying to measure the Hubble constant.

I remember from a cosmology class (can't find a source at the moment) that the Coma cluster is the first cluster whose redshift is dominated by the Hubble flow as opposed to peculiar motion. This is mainly due to the fact that the local group is moving towards the nearer Virgo cluster. From the above wikipedia link: Distance ca. $100 \textrm{MPc}$, apparent recession velocity ca. $7000 \textrm{km s}^{-1}$. So, at that speed, you would definitely see a systematic difference in redshift of galaxies ahead of and behind you. How long to get to that speed at constant acceleration of $10 \frac{m}{s^2}$?

\begin{equation}
 a = \frac{v}{t} \Rightarrow t = \frac{v}{a} = \frac{7\cdot 10^6 m s^{-1}}{10 m s^{-2}} = 7 \cdot 10^5 s \simeq 8\, \textrm{days.}
\end{equation}
Note that this is a worst-case scenario. People have used much closer galaxies to determine the Hubble constant. This paper's abstract says 


  Due to the linearity of the expansion field the Hubble constant $\textrm{H}_0$ can be found at any distance $&gt;4.5$ Mpc. 


This is all very local, so we'll just assume a Hubble constant of $\textrm{H}_0 = 70\textrm{km}\,\textrm{s}^{-1}\,\textrm{MPc}^{-1}$ (Planck says 67.8 and Riess et al. say 73.8) and calculate redshift and hence apparent recession velocity at that distance using the Hubble law

\begin{equation}
 v = \textrm{H}_0 \cdot d = 70\,\textrm{km}\,\textrm{s}^{-1}\,\textrm{MPc}^{-1} \cdot 4.5\, \textrm{MPc} = 315\,\textrm{km}\,\textrm{s}^{-1}\,.
\end{equation}
At constant acceleration of 1g you'd get to that speed in about 9 hours, but you might need more time to do your data analysis on the galaxy spectra you'll have to take.