Once you get past your ex-wife's sniffy remarks about making sure you're sober as you exercise your visitation rights, being a non-custodial father is not all bad.  
After all, there are no more diapers to change, no more sidewalks to shovel, and (above all) no more having to look at the back of her head as she turns away from you (again).
Besides, if you are reasonably presentable, reasonably articulate and 
reasonably attentive, there are so many willing, good-looking women out there, especially the older ones.  And the majority of the older ones are so grateful, unlike you-know-who.  I might add you should stay away from the younger ones entirely.
Sometime after you split up with Her, you will discover that the singles scene is nothing like the dating scene that you (and she) left behind so long ago. I have written two essays on the subject to help you along in this regard.  The first essay is entitled "Pee Wee's Excellent Adventure: DWM, Age 52, Has Own Hair, Own Teeth; Hopes You Do Too".  My readers especially liked the list of do's-and-dont's in that one. The other is entitled "Pee Wee:  Life in the Fast Lane".  Neither of my essays will ever be nominated for the Pulitzer Prize, but they do have a proper place on the shelf.  They are available on request.
In preparation for your first visitation with your kids, please remember that the alimony you have to pay your ex may in fact be negotiable, but the child support is a sine qua non for domestic tranquility.  Not paying child support is like walking around with a lightning rod on a stormy day. Therefore, if you fall behind at all, you should immediately take all reasonable steps necessary to either: (1) be actually current on your child support payments or (2) at least have the appearance of being current.
Actually being current is not nearly as much fun as it might seem, because arranging to be current can be such a pain. On the other hand, achieving the appearance of being current is an opportunity for personal growth and affirmation that should not be lightly passed up.    
Let me explain.  Over time, you will doubtless observe that the level of 
verbal small arms fire directed at your person during visitation is usually directly proportional to the level of your child support arrearage.  This is usually expressed as SAF:CSA = 1:1. If you don't believe this proposition, you are invited to try it.  However, please don't just jump into the testing; do it gradually, testing and refining a little bit at a time.  If in fact if you are a sufficiently diligent and attentive student, your "arrearage sensitivity" should become well-calibrated over time.  To put it another way, you will be able to 
read and play the score like a master pianist.  After some practice, you might be able to massage the usual cacaphony of  small arms fire into something of a symphonic presentation.  If you become very very good at it, you may even find your very existence being validated by the experience.
So, if you know you are going to be behind at all in your child support to any appreciable degree, it would be a excellent form of damage control to send your ex-spouse a copy of your paystub to show how little you actually earn. This can also be regarded as a form of "dialogue".  As your friend Bubba might say, "There's more talkin', less squawkin', so to speak".This grand gesture will operate to calm her down, at least to some extent. Later on, your paystub will be referred to as "Defendant's Exhibit A".
If she looks over your paystub and still thinks you have too much disposable income, seize the initiative immediately.  Look her straight in the face with a deadpan expression of earnestness and offer to move back into the former marital premises, so you can pay the rent directly to her.  "Think of all the money you'll make", she can be told. " And you can keep an eye on my lifestyle expenses, too."  Her boyfriend will choke at the suggestion, much to your satisfaction.  If you were in fact the abusive alcoholic airhead she always said you were, she will never agree.  Or, if she is in fact the mindless tool of that gossiping gaggle of girlfriends you always said she was, she will never agree because her girlfriends will never let her.  
In other words, the offer to move back in to save money is one of those grand gestures you'll never have to make good on.  And, if you really need an edge in the courtroom sometime, you can mention your offer to the judge, to make sure he or she knows that you were willing to walk back through the Jaws of Hell for your kids.
Some non-custodials have actually gotten a second job at low pay just so they could show the second paystub to their ex-wife and keep their main paystub  to themselves.  I don't recommend doing this, if only because the nighttime work involved in the second job is such a pain.  Besides, it will interfere with your love life and your love life is why you got the divorce in the first place.
In any case, sending her your paystub will help to maintain diplomatic 
relations even during the bad times.  It will show that you are trying ("Yes, very trying," your ex may say, but she will still appreciate the paystub).  Remember at all times that you have to get past her to get to the kids.
When you show up to exercise your visitation, it helps to show up on time.  If you show up late or not at all, she will get the impression (again) that you are playing mind games (again).  You absolutely know that her giggling gaggle of girlfriends will use their own agendas to analyze your lateness and they will conclude that you're playing games (again). 
It will sometimes actually happen --  through manifest misfortune and not through neglect  --  that you are late in arriving for your visitation.  Your ex will consequently be obnoxious as you walk in through the door.  My advice is to admit nothing and to deny everything.  In fact, I suggest that the only thing you can possibly do under the circumstances is to ignore her mood in as obvious a way as you can.  
In other words, greet her with a warm handshake and with a toothy, expansive smile on your face.  She will know that you know that she knows and it will absolutely infuriate her.  You were married long enough to her to know when she gets heartburn, and this will be one of those times.  It will be immensely satisfying.  You will have a taken a lemon of a situation (lateness for you) and turned it into lemonade (heartburn for her).
If you have no doubt that she is going to be obnoxious because of your arrearage or because of your lateness, arrange to have just a soupcon of your girlfriend's perfume on your chin.  The exact amount of perfume has to be carefully calibrated, because if it is too noticeable, she will be onto your game and she will get even, probably by wearing someone else's Old Spice, which you know will make you go ballistic.
Besides, if your girlfriend's perfume is too noticeable, the kids will pick up on it and be totally grossed out.  Once the kids get grossed out, she will maximize the situation and try to score points on other issues, such as your character, in front of the kids.  In any case, the perfume will become  one of those tawdry little details that she will cheerfully make sure the judge gets to know about.  She might even put the kids on the stand, which would be intolerable.  And she will make sure the judge knows it is your fault that the kids have to be put on the stand. Watching your own kids being put under oath is not as much fun as it might otherwise seem. 
After you have arrived on time, poor and sober, make sure you have brought the most important thing for your kids: gifts.  You may think that the kids can't wait to see you, but they actually can't wait to see the gifts you brought.  You, personally, are negotiable.  Looking at you, the kids will say to themselves, "You gotta play the hand you're dealt", meaning you, the original busted hand.
The gifts cannot be anything showwy, because Herself will be calculating the price tags, and wondering out loud if you have too much disposable income.  Also, the kids will figure out that you're trying to buy their attention, which of course you are.  Making it obvious, however, presents the kids with ethical questions that they simply do not wish to address.  They might have to actually think about ethics.  Heck, they might even be forced to develop some character, the poor devils.
Unless there is a court order to the contrary, you will not be exercising your visitation on your ex-wife's premises.  This means you will have to take the kids somewhere, hopefully somewhere not too expensive.  Your ex-wife has made it as clear as the bells in the Kremlin that "somewhere not too expensive" does NOT include your girlfriend's house.
Then, after all this, I suggest you take your kids for a walk in the park.  
All of you should take time to smell the roses at the edge of the pathway.  Certainly, roses do not have quite the panache of Old Spice or Arpege, but they’ll have to do.
As the kids might say, “ Ya gotta smell the flowers you’re handed.”