First off, there is no way you can be a failure as a Dad when you are taking your kids fishing. My best memories hands down as a kid were fishing with my Dad and his buddies. Good on you.
Do you guys have a Cabelas near you? Their brand of spinning gear is actually quite good, especially for the price. You can generally find a combo on sale for cheap. I also find their staff to be knowledgeable and won't oversell you. They don't own the business, they just work there and love to fish. Not sure what your budget is but for entry level you could find a serviceable combination for 50-75 bucks. Just look for something on sale, don't pay the full price, and this is the perfect time of year for that.
If Cabelas isn't an option, I am not familiar with Bass Pro but I am sure they have a similar set up.. but I won't be able to speak to their quality. Don't over think it for the basic stuff, just look for what is affordable. Spend money on good line though, most places will put it on for you right there. You can avoid a lot of tangles with a decent set of line. If you buy from CT especially make sure you take off whatever crap they have on there already.
For knots I always use a Palomar, once you get it down it is the fastest knot to tie and your kids will not be losing any fish with this one.
http://www.animatedknots.com/palomar/#ScrollPoint
Edit: That's spinning and bait cast, stay away from the bait casters those are generally more expensive and are better saved for an experienced angler. Kids will do fine with the spinning reels, just teach them to get the timing right on the release and like I said good line will take care of a lot of your tangle problems.
Edit edit: If you're going to CT and Cabelas or Bass Pro is an option definitely stop into one of those first before you buy, I honestly think you will find the prices to be very comparable for better quality. Like I said, it is sale season.