Wego is right, for the peonies either spring or fall will do, and fall is best. For the Iris, the best time to split and transplant them is 6-8 weeks after they finish blooming. The following are detailed instructions for splitting and transplanting both plants.
When splitting your peonies, the most common major mistake people make is to plant them too deep. Be sure to plant the division at the right depth--plant them too deep and they won't bloom. Also, it generally takes two to three years for them to begin blooming after transplanting, so if they don't bloom for the next two or three years, that is normal. Peonies develop an extensive root system that can take a couple of years to establish. It’s not unusual for blooming to be sporadic or delayed until year three or four.
Peonies are long-lived plants--many will live more than 50 years--so try to plant them in a permanent location where you won't want to move them. Also, they like reasonably well-drained soil and need a minimum of 6 hours of direct sunlight per day.
Some things to know/do to successfully divide peonies:
1. Take care to avoid or reduce injury to the roots when dividing or transplanting peonies.
2. To begin the process, start by moistening the soil around the plant several days before lifting.
3. Cut the soil around the crown to a depth of about 8 inches and then lift out the crown.
4. Wash the soil from the roots, cut the stalks off within 1 inch of the tip buds, and with a sharp knife or spade, separate the clump into several divisions containing three to five buds each. The growth buds are the dark red “eyes” at the top of the roots
5. Crowns and roots will vary in size and shape, but buds and roots must be in each division. The most vigorous portion of the crown is usually found around the crown’s outer edge.
6. Next, move the divisions to the prepared transplanting site and set the divisions shallowly, with buds no more than 1 to 2 inches below the soil surface. Planting too deeply can result in plants with abnormal growth, reduced vigor and little or no flowering.
7. Fill soil between the roots so that there are no air pockets, then water thoroughly.
8. Apply an organic mulch 2 or 3 inches deep to prevent winter heaving the first season. After the first year, mulching is usually not necessary for herbaceous peonies. Keep newly planted peonies evenly moist.
You should understand the Iris life cycle--each individual rhizome will only produce one flower stalk during its lifetime. After (and sometimes before) it flowers, it will turn its energy toward producing "increases" (new rhizomes growing from the sides of the "mother" rhizome). After these new rhizome grow to blooming size and eventually flower, they will then become "mothers" and grow increases of their own. That is how single rhizome turn into clumps over the course of a few years and why they need to be divided when the clumps become overcrowded and deplete soil nutrients.
In regards to your dividing and transplanting your Iris, the following may be helpful:
1. Don't transplant irises in very hot weather (90+ F degrees), but DO make sure to get them transplanted back into the ground a couple months before freezing weather to give them time to root in well before winter.
2. Iris sometimes take a year off bloom after transplantation to settle into their new homes, so don't be disappointed if they don't bloom the spring after planting. Irises moved before bloom will often lose their bloom that season as well, which is why it's best to wait until after bloom to dig and divide.
3. Use a spade or fork to dig up your iris clumps and wash them off until you have removed all dirt and can easily see where the rhizome are attached to one another. With a sharp knife, cut through the attachments (you can break them apart with your hands, but a sharp knife will make a smaller, cleaner wound).
4. Inspect the rhizome thoroughly. Discard any which are soft or mushy or show any other signs of disease, any that do not have viable roots, and any that are just too tiny to bother with. Many people also discard old "mother" rhizome that have already bloomed because they will not bloom again. These "mothers" may grow additional rhizome (increases) however, so they may be worth re-planting if you have the extra room for them.
5. Remove any dead, spotted, or unhealthy (brown or yellow) leaves or parts of leaves. Many folks cut the leaf fan back a bit when transplanting so the weight of the leaves won't cause the newly planted iris to tip over, but you should leave as much leaf on the plant as possible to continue to feed the rhizome. Snap or cut spent bloom stalks off at the rhizome. Snip off any dead roots but leave the plump roots intact.
6. After you have separated, inspected, and cleaned up your rhizome, lay all of them out in a shady, dry area for a couple days to allow the cut wounds to scab over before transplanting.
7. Transplant these Iris divisions into an area providing at least 6 hrs of direct sunlight a day, making sure the soil is well-drained and properly fertilized. Avoid high nitrogen fertilizers. Excessive nitrogen can promote bacterial soft rot in bearded irises.
8. When you plant your irises, make sure the top surface of the rhizome is level with or just slightly below the soil surface. If you bury the rhizome too deeply, the plant may refuse to flower.
9. Do NOT mulch directly over the rhizome. Mulch will tend to retain too much soil moisture right around the rhizome and promote bacterial soft rot (unless you live in a hot desert climate). If you have the room, plant your rhizome about 2' apart with no other plants nearby to overgrow them and compete for soil nutrients. You can also plant 3 separate rhizome in a triangle 1' apart with the leaf fans facing outwards to produce an instant clump effect. Water the irises deeply once a week for the first month if there is not adequate rainfall. Avoid frequent, shallow waterings. Over-watering is another common cause of soft rot problems.
Good luck with your gardening!