Your family is fine for camping. My concern would be your 10k budget and your Pilot. MH’s tend to be money pits; you could spend $2000 just replacing tires. And you’re probably going to throw a LOT of money at an old one. Plus your Pilot doesn’t have the capacity to pull much of anything.
The DW has been promoting a Class B for traveling and I went into sticker shock when I saw what they cost. A C is cheaper, but they’re not cheap, either. There might be some really small, light units, like teardrops, out there that you could pull. But with no experience with them I don’t know how they’d be for 2 adults and 2 kids. Coleman used to make lightweight, all metal PU’s that might work, but they’re gone. The newer ones I’ve seen outfitted with showers weigh more than my 16’ TT.
The advice to take your time and rent is good. There are all types of RV’s and each have their pros and cons and you need to educate yourself well before you jump into anything. There are a lot of posts on the forum about what sort of RV to select and you can do a search and get a quick feel for what appeals to you. Good luck.
