First, I think some of the earlier posts gave good input. Get them and yourself on a good schedule. That doesn't mean you have to schedule tons of activity, just a basic schedule. Make it suite your family's needs and desires. Something like an hour getting up and ready in the morning, have breakfast together, play for an hour, put on TV for 1/2 hour while you get a bit of work done, let kids play in their room for an hour while you work on other stuff, make lunch, go to a playground (weather permitting) eat then you and kids run off some energy, run errands with the kids unless someone agrees to watch them while you go yourself, go home put kids down for a rest (even if they won't go to sleep), work on the house, dinner, read a book or take a nap yourself while they are napping, after they wake up play a game with them, do an activity, or work on educational stuff for about an hour, have dinner, then start winding down for the night with a regular nighttime routine. It can be whatever suits you, but keeping a regular schedule can help move things along, avoid the trouble of figuring out what to do next and finding the time to do what is needed, get the kids used to a given set of events, and limit some of the tantrums you talked about.
A daycare for 1/2 days even if only a couple days/week can be a big boost. It allows great social time for the kids, and they get more used to playing with others because they are around lots of kids on a regular schedule with people directing activities. The time they are away gives you a perfect time to relax for a while, get some work done, take a breather, and maybe even enjoy your own time for a while. Just be sure to do your homework and find a preschool reasonably priced with a good environment, good teachers, and a good schedule. The preschools are really no more expensive then just daycare, and most daycares operate like preschools anyway, even if they aren't labeled as such. Visit them a couple times before deciding, and make sure you bring the child at least once so you can get a good idea of how they feel about each one. I let my daughter help me decide on a facility by asking her which ones she liked and why she liked them. I had to turn down one because it didn't seem safe, and another cause it was too far, but we both agreed on 2 facilities and ended up deciding on the one with a more flexible schedule.
I deal with the TV issue myself. I usually just decide for myself when my kids ask, if I want it on or not, based on how long, what we are doing or getting ready to do, and how much has already been watched. If my son had his way it would be on all the time, but I don't allow that. I don't care if he has a tantrum, once I say no, that's it. Don't say no and then change your mind because she's crying, begging, or screaming. That will encourage her to do it every time you don't go along with what she wants. Sometimes I cut out TV all together just to get them used to not having it for a while, and it starts off with lots of whining but it ends up more peaceful, and we do more together. There is a set of programs I like to put on some nights that helps my kids get to sleep and that is the night time programs on Baby First TV. They play soft music while putting calming pictures on the screen, I even use it myself sometimes to help me get to sleep. It doesn't start that early so I record some of it for the kids to get to sleep at a good hour.
The trouble with the fighting, crying, and nagging is that it is perfectly normal for little ones. They will do that even if you are a perfect parent, and quite frankly, no one is perfect. Trying to appease and entertain them to avoid these things doesn't work, and probably makes it worse because it puts the little ones in charge instead of the other way around. That doesn't mean you need to require them to only do what you want, but there needs to be some give and take. Keep them involved in decisions that involve them, but make sure you have your own input too. The entire family has needs that must be met, not just the kids, and not just the parents. Try to include them while still including mommy's and daddy's needs. Doing activity together is good for the family, and their development, but it shouldn't be done with the goal of entertaining the kids to avoid tantrums and fighting. It should be done with the goal of having fun, and having an active involved family life. The kids will inevitably fight, because that's one way they deal with their frustrations. Rather than trying to avoid all fighting and tantrums maybe it would be better to allow them to fight once in a while, but try to guide them to work things out. Don't give to much guidance, just little nudges like saying "I see you are having some trouble, is there a way you can work on it with ..., what can you do to....how can she..." just little interjections when they get stuck, but only enough to aim them in the right direction. This allows them to learn how to deal with their problems, and to compromise, rather than making them into your problems to solve for them. Another thing to try is giving them little lessons on anger management, like taking a few slow breaths, counting, squeezing a squishy ball, walking away from the problem for a while, doing something else that isn't so frustrating, or taking a time out. I don't mean the traditional time out in a chair or corner. I have an area in one rarely used room that I let my kids set up with a couple bean bags, blankets, pillows, and books, that is their quite area where they go when they want a little time for themselves, that is what I mean buy time out. The only problem they have had with it is that some times one of them wants to be there buy themselves, but the other ones wants to be there too. Usually it works well, sometimes they even sit there together and look at books together, which is truly a godsend.
Oh one other thing. With the housework, I find it very helpful to involve my kids. I usually give them a very simple task that won't be a bigee if they mess up, like putting dishes in the rack, or wiping down the table with a sponge or rag. If I have them do something a little easier to mess up, I just give them more guidance. They don't always give a lot of help, sometimes they barely help at all, but what this does is teach them how to be productive and helpful, while keeping them from making more messes while you are cleaning. So even if they don't do much, they are not causing a hurricane behind you while you clean.
I hope that wasn't too much, and I hope it is helpful. Good luck to you, and remember not to be too hard on yourself. The house doesn't have to be perfect, the kids don't have to be perfect, and the mommy and daddy don't have to be perfect either. Just as long as you keep things safe and fairly comfortable, make sure there are clean dishes and clothes, and healthy food, you and your family should be fine.