The simpliest way to make this as easy as possible is to put your own wants aside and decide what is best for the children. To transfer them from one home to another at bed time on a school night may be the most convenient for you, but it's not going to help them stay on schedule.
Speaking of schedule...it is so important to keep them on a similar schedule and routine as much as possible. Once you get a child into a routine everything with them is easier. A steady night time routine gets them into bed and asleep faster, a morning routine can get you out the door on time, and keeping bed times and wake up times the same will maximize their sleep and keep them less cranky at both times. Sure, this takes a little communication, but it's not as hard as you'd think.
When they are young they often have one item they are attached too. This is a GOOD thing. It offers them consistancy when everything is changing and gives them a sense of security, that not everything is different all the time. It doesn't matter who gave it to them or why, but removing this item while they are at one house or the other can make it drastically harder for them to adjust. Now not only is everything different, but their security item is gone, so they just don't always know what's happening or what's left. Young children don't understand why they can't have their favorite toy at your house, so just let them.
Respect the other parent and be on time. There are situations where things come up, yes, and some can be preplanned while others (traffic jams) just happen. By being on time every other time these won't seem such a big deal, and the other parent will not be angry or resentful. It also shows your lack of respect to the parent as well as the court order (please always have it court approved and in writing...) and therefore the court itself. My ex was able to learn first hand last year just how the court feels about you disrepecting them and their orders, while I found it slightly amusing to finally see something said about it, his red face and embarassed reaction made me feel it was unpleasant for him. Please also note that breaking a court order, especially repeatedly, is considered contempt of court...
Communication is key. Now I know that we all learned that about being together, but it is also extremly important now. If they are sick both parents need to know what it is, how long they've had it, and if they've seen a doctor. You need to decide who takes them for flu shots, when they need to see a dentist, and if they have their eyes checked. Don't forget injuries! These are VERY important to share information on. All kids get hurt, but if not all the details are known they may not receive treatment they need, or symptoms may be missed as you won't be looking for them. This can be dangerous for a child and parents may be charged with neglect in some circumstances.
Allow them to have things that they take with them to both places. Children have favorite clothes, toys, and books. It's hard enough to be away from a parent but to also have to adjust to another full set of things they aren't attached to or have memories of will only make it harder. Taking them somewhere? Take their favorite jacket, or in some cases only jacket, so they know that it's familiar and they associate it with safe and happy memories, not new and scary. It also makes it easeir for them to recognize and keep track of their own items.
Attempt to have similar rules and diciplines. Consistancy ensures they know what to expect that they will be better able to follow the rules if they know what they are. It is hard for children to learn boundaries and expectations, and when these differ from day to day it's even harder. I know that not all rules can be the same, and that every person has different views on dicipline, but this is not about the parent.s, it's about the child.
When you have time with your child, try and spend as much of it actually with them as you can. Yes, people have to work, but if you only have your child one weekend out of two, you don't need to go out the whole weekend and leave them with a sitter. They do need to see both families, but to take your child and drop them at a family members house for 3 of your 4 days is only going to confuse them more, and perhaps cause them to feel that you don't want them around. Children also remember more then we think, and I promise when they are older they will remember you took time away from one parent and didn't even want to actually be with them, therefre they will likely conclude that you did it just to hurt the other parent, or even themselves, and this WILL negatively affect your relationship. They do need to learn that other people care for them and that other adults can look after them and keep them safe, but you need to put in a decent amount of time to show them they mean something to you, that they are important.
Try not to fight in front of the children. I had this problem with my ex a lot, and he was very rude and angry with me over what I felt where small issues, or things that could be solved easily as adults, but yelling and screaming will only frighten and damage the children. There have actually been studies showing that that behaviour will physically affect the development of a child's brain.
Be positive about the time they spend with the other parent. Children internalize a significant amount, sometimes everything, and they will feel guilty about wanting to spend time with the other parent as it is time away from you. If you can show them that you value their relationship with the other parent they will feel less like they are hurting you and maybe enjoy their time a little more. It is also damaging to your relationship with your child if you create difficulty between them and the other parent, or make the child feel guilty, they will grow to resent that later.
I think that the easiest way to navigate this is to remember that it needs to be about the CHILD and their best interest, and parents come second. That's just how life is at those times, and there's no way around it. If you can put what is honestly going to help the child the most in front of what you want then they will adjust quicker and better.
Please feel free to comment with any of your suggestions or what you may have learned.