Not sure what cameras you have had before but the AF system on D800 is quite complex, what with choices between AF-ON activation or shutter release activation, AF-S and AF-C, single and multiple focus point selection etc.
It is worth reading up on it and setting the AF systems on your various banks of settings to give the responses required for different types of shots.
I agree with all the comments about high resolution and higher shutter speed. I have found this in practice myself after moving from DX cameras. Be aware also that FX format reduces your depth of field (relative to the wider field of view), To get the same effective depth of field as you might be used to from a DX camera, you need to shoot up to 1 f-stop smaller.
To see if your hand holding technique is the problem, try shooting a few pics as you normally would, follow with more by using successively faster shutter speeds each time. If you notice a significant difference between fast and slow speeds - your handholding technique may require improvement - or some additional support.
If your AF system is seriously malfunctioning - try mounting camera on a tripod - take a few shots using AF and then the same shots using Live View to focus. the normal AF uses the focusing screen under the SLR pentaprism while Live view uses the actual sensor. Again if you notice significant differences then your AF system may be faulty but if there is only a marginal difference then AF system is not likely to be the cause.
Finally, if it is clear to you that the lens is consistently focusing in front of (or behind) your subject on a regular basis then AF fine-tuning may be needed.