Hi, Welcome and glad to hear you are interested in learning to paddle.
What you have there is called an inflatable kayak. I have no idea about the specifics of the model that you have, as it clearly predates my entry to the sport
There is a lot you probably need to know before you actually take your family out on the water. For your saftey and theirs its best to be informed. There are the basics. Make sure that everyone in the family is equipped with an appropriate life jacket. For slow moving water the Coast Guard Approved Class III lifejackets available at many retailers are just fine. Make sure you know how to properly wear them.
Please be sure to carefully research any stretch of water you intend to paddle. Unless you are 100% sure (preferably have already been there) about what you are getting into, you have no business taking your kids. The most common and tragic accidents happen when well meaning parents take their kids on fun 'rafting' day trips without understanding the consequences of what they are getting into.
As always, when you are out of doors, plan for changes in weather and that goes double with kids. Things can go wrong, your boat can sink, storms could happen... have a plan for how you handle those situations at a minimum.
Okay, now that I have preached a little... to answer your specific questions. (BTW, there is tons more to know before you actually get on the water with this thing, but the above is just a sample).
No, you don't need a frame. That boat is designed for one to two adults to sit in it and paddle it down the river. You will want to have at least one of two inflatable kayak paddles. A simple Google search of this term will give you plenty of cheap options ($25 or so). The paddles separate in the middle so that each person can paddle with a double bladed or single bladed paddle. Typically this boat is designed for about 400-500lbs, so if you are taking your family with you be sure to keep that in mind.
Keep asking lots of questions from anyone you know who has paddling experience, and good luck! Be safe, and have fun!