Welcome. It's great that you're open to learning, makes you more attractive. Anyway, that's the key - to be more attractive. I don't mean in looks necessarily, although you should definitely make sure you look wonderful every time you see her - buy some better clothes, get a good haircut, get a manicure - those things do matter to women.
Outside of that, though, you can become more attractive by learning and changing and being open to ideas; at the picnic, ask her honestly what you did wrong. Tell her that you're ready to 'fix' yourself - not for her, but for you. You need to know what you did that caused the divorce. For now, even if you think she was at fault, don't say so. Just listen. Let her get it all off her chest. Then take that knowledge (those things you 'did' are called Love Busters), and stop doing them! Make a conscious effort to catch yourself doing them and stop it.
Also ask her what she would have wanted you to do, in a perfect world. Those things she wanted from you are called Emotional Needs here at MB. Let her spill her guts to you, and learn what it was she really wanted. Then find ways to become that one person who knows what she wants and knows how to fill those ENs. If she says 'I needed a husband who kept the house in shape, and you never did,' then every time you visit the kids, look for one thing you can fix - and say you're doing it for her and the kids because you'll never stop loving them all. Let her see you in action, doing those things she said she wanted. But don't be obvious that you're wanting a response from her. It has to be that you're doing it because you're growing as a person, not just to win her back; that you would be doing this for anyone you cared about.
Finally, at least for starters, take a big interest in the kids. Don't just be a weekend dad. Find every way you can to be involved in their lives. Attend every dance recital, every baseball game, every event you're allowed to attend. Shows her you're good dad/husband material, and it will make a world of a difference to your kids, having to deal with divorce.