what is the best time of year to move bushes and shrubs

Anything you wish to plant or move can be undertaken from October to March, if the ground has been prepared, is not waterlogged or frozen. This is the general rule.

Bare rooted shrubs and trees are most safely moved in late autumn or winter during their dormant period.

as above. make the planting hole bigger than the space they are coming from. keep as much soil around them as poss. dig in compost/leafmould into the hole they are going in. you will ahve to keep an eye on the watering and depending on what you are moving you may have to reduce the leaf area by pruning, to reduce waterloss/stress.
what are you planning to move?

Don't forget to take into consideration whether the plants will be in similar light as before, ie, full sun, semi-shade etc, otherwise they might die despite all your tlc.

Also, if it's a very large and established shrub I would seriously consider moving it over two years. First year, dig a trench all roound and cut back some of the the longest roots which stick out beyond this. Then refill with good soil. Over the next year, it will compensate for the loss of these very old and established roots by growing new, smaller ones and making a neater rootball. This can then be lifted and moved the next year with a much better chance of survival.