My advice is to think about everything before you decide to date. Regardless of it all, you should never bring anyone around your children, until AFTER you have taken your time to get to know them first, I would even say maybe not bring the person around unless the relationship is truly turning serious, and the possibility if marriage is coming. There is a lot to think about in this situation. Your children could react to someone you are dating in many ways... they could reject them, give them a hard time, think you are replacing their mom/dad or etc... if any of these are the cases, then it could damage your relationship with the person you are seeing, AND/OR damage the relationship you have with your children, so you REALLY need to know that this person is in it for the long haul, and not going to be gone next week. Another thing about this is if it is a NEW relationship, and maybe you have only known the person for like a week or something, if they are suddenly faced with your children, and the children are reacting badly to them it could be more than they can take, and might not think you or your kids are worth it, since hey they just met you, and there is no love in their hearts yet or etc... However if you have a person you have been dating for a while, have taken the time to get to know, and there is "love" between you two, that person is going to be more invested in you, your life, come good OR bad, which means most likely when they meet the kids, if the kids react badly they will be more likely to be able to help you and them work through it, more understanding, accepting and etc... in this case you have a much better chance of everything working out.
Not all children will react badly though, and this can be a good thing but in many ways it can be a very bad thing. What I mean by this is... if you bring someone around and your children except them right away, enjoy having time with the, and react well this is great, however, if this person doesn't turn out to STAY in your life, and they are gone the next week or month, it is going to have a BIG and bad impact on your children. In many cases it would be worse damage on them in the long run. If your children are sensitive, excepting, and etc... you really need to be careful of ANYONE you bring around them, even if it is just friends, they don't even need to be someone you are dating, it can be anyone the children might attach too.
I personally would say you should never bring anyone to your house, until enough time has passed, and you and the person you are dating have gotten serious, and committed to each other. Even at this point, I would say start by maybe having dinner out with the kids there, or go to some place like chuck e cheese, with the kids. Give them chances to meet this person as your FRIEND, in a safe environment, that is not a threat to their "personal" space or home. After some time passes, and there have been a few interactions with them outside the house, then maybe have them over during the day, maybe to watch a movie or have dinner with the kids there. Just progress slowly, and handle any reactions the children have with understanding and compassion. Remember some will think you are trying to replace their mom/dad, some might become fearful that they will "lose you" to the person you are dating, or etc...
Most importantly, you will be teaching your children a LOT of things in this time. Do you really want your children to grow up and develop into a person who sleep with anyone and everyone? Someone who goes through relationships like toilet paper? Children learn by example so remember this.
Now two more things, which no one has mentioned yet...
If you are bringing in different men/women each week, or month, and word about this gets back to your ex, you could have the possible problem of the ex seeking custody based on this. Depending on how your ex is, you could risk losing your children because you are thinking about your libido more than your children, or the impact your sex life is having on them. It could easily be used against you in court or etc... Also depending on where you live, like if you live in a small area or town... if you date a lot of different people, or bring them home, word could get around that you are a "*bleep*" or etc... and this could have an impact on your children, and their lives like in school, or around their peers.
The second thing no one has mentioned, and possibly they VERY most important thing... you NEVER know who someone is... remember the person you are dating is NOT the biological parent. They could be an abuser, sexually, physically, or mentally. You could risk your child's mental well being, physical well being or sexual well being. I am dead serious about this, in this day and age... do not be so gullable to meet some man OR woman, and after a week of being woo'd think "oh this is a great man" and bring him around your children, only to find out later he is a child molester, or he drinks more than you thought originally, or etc... I don't care how nice the person seems when you meet them, they CAN be any of these things. Most people hide who they are on many levels from someone they are dating. It could be that they are hiding the fact they chew their nails, have stinky feet, fart a lot, or etc... or they can be hiding something much darker. TAKE the time to find out what if anything they are hiding, take the time to get to know them, learn about them, meet their friends, family and etc... NO amount of sex is worth ruining your child's life. NO man or woman is worth it.
PROTECT YOUR CHILDREN, instead of just FEEDING YOUR LIBIDO, or FILLING AN EMPTY SPOT LEFT BY YOUR EX.