You didn't say what level of paddling you and your wife would be looking for; but we've got the range from flat water "swamp trips" as we call them, to mild WW, to some pretty good WW that can get up to the IV range, +/- depending on levels, although catching those in June would require a lot of luck with the rain. Here's my list, and why I like each one:
Hiking:
1) Pinnacle Mountain (go to the top for exercise and the view). Close and very scenic.
2) Petit Jean - Hike to the base of Cedar Falls. Several other nice short hikes too.
3) Lost Valley, near Ponca, Upper Buffalo Area - Very scenic trail; an Indian Rock House (overhang bluff), but for a real treat take a flashlight and crawl into the cave where the creek comes out of the mountain. There's a beautiful little waterfall in there a short distance back. About half way up the trail to that cave, the creek also goes through a short cave that is too long to be called a natural bridge, but maybe too short to really be called a cave. You don't need a flashlight for that one. The creek through this one has been run at high water, a trip I regret missing to this day because it cost me not one, but two or three "first known descents" in a single day. I was in route to join those who did it, and my pager literally went off as I was driving past my place of work. (I was foolishly responsible, hoping it was only a simple question I could answer as I continued driving.) That memory helped me decide to take an earlier retirement.
4) Goat Bluff trail, also near Ponca. Great View overlooking the Buffalo River, don't fall!
5) Pedestal Rocks, just east of Sand Gap/ Pelsor, which is North of Russellville on Hwy 7.
Paddling:
1) Steel Creek to Kyles (some still put in upstream at the Ponca Low Water bridge). Many Arkansas Paddlers trace their very first moving water trips to this specific stretch of the Buffalo. Along the way, if you don't stop and take the short hike in to see Hemmed In Hollow your first time, you should be tarred and feathered. Totally suitable for beginning paddlers in the company of others. Often hard to catch with enough water by June, but if you get there and the water is too low, two of the hikes mentioned above are within 10 miles. You can of course check water levels before you go, the National Park web page has links to gages.
2) Rockport, with guaranteed dam releases every Saturday and Sundy, great for fun, meeting other paddlers, and "park and play" skill building, near Malvern (45 minutes from LR if you drive slow). Suitable for all skill levels; there are easy surfing waves, and some hydraulics where kayakers in play boats work on more advanced skills. There's also a lazy downriver float that ends at the play spot, most Saturday's someone does that so you can watch the Trips Forum and find folks to shuttle with.
3) Big Piney Creek if it runs in June (North of Russellville)
4) Cadron Creek if it runs in June (Just North of Conway)
5) Cossatot River (western Arkansas) - if you have Class III /IV skills - but this one is a long shot in June. If you're an advanced boater this should be on your "must do" list, but travel with other experienced boaters is a REALLY REALLY GOOD IDEA for this one.
6) Swamp Float - Little Maumelle - Put in at Pinnacle Mtn State park boat ramp; paddle downstream, drag one or two shoals if necessary, keep going downstream. Beautiful Scenes and some birding. Return to your starting point. Through floats are also possible if you have two cars.
7) Swamp Float - Watttensaw Bayou Water Trail - 45 minutes East of Little Rock. Can be paddled all summer; info on Game and Fish site, again can be enjoyed by paddling "in and out", or via through floats if you have two cars. There will likely be one remote float camp in place by June; my wife is committed to finish that SOON. Big trees; birds and other wildlife. One of the cypress trees is so big that we can squeeze through a hole and get INSIDE the tree; I've been in there with up to three other people all standing around with flashlights. Bug spray needed around sunset in June, but during the day it is normally not an issue except on cloudy days.
Checking levels on many of these things - find the links to either the AWA page or the Ozark WW page from the front page of this site,
http://www.Arkansascanoeclub.com" onclick="window.open(this.href);return false;. Those are both coded to show "too low", "low", "good", and "getting too high". Terminology differs for what you call these levels, but "too low" generally really means you'd be hiking, too high means you'd need skills well above average for that type of water if you want to stay safe. The swamp floats I listed are not on our levels page but I've pointed you to sections that should be good all summer.
Gluten Free eating:
Big Orange in Little Rock - has some local brews (not gluten free of course), but also has a gluten free burger bun that I actually liked better than their regular bun. (I can eat either but I was just curious.)
Additional: Find Elene Murray on Facebook; she would know all about other gluten Free options.