First holiday where to look where to go?

Your budget sounds fine to me - I usually go away for 2 weeks for less than that. Are you looking at a hotel with breakfast included, or self catering?
Greek Island s/c tends not to have pools although you can often use one at a nearby hotel.
If you go to a travel agent you don't have to buy on the spot. I often go and ask what they've got, then and have a coffee somewhere while I check out the suggestions online.
I don't pay extra for sitting together - if the flights only a couple of hours I don't see the point. If you book a package from a high street travel agent a case each and transfers should be included. Don't buy the travel agent insurance, you can get cheaper elsewhere.

My partner and I have been on loads of holidays. Most recent was Croatia which was wonderful (10 days October, from Dubrovnik split hvar 400 each) lovely weather, we stayed in apartments from air Bnb etc and did our own flights, ferries etc. Loved the multi centre holiday and hvar is just beautiful, Dubrovnik has the history plus beaches and islands nearby. This was eleven the migration issue was at the fire but we saw nothing of it.
Majorca is beautiful but full of kids, albufeira in Portugal is hot yet breezy and has a very warm sea and a new and old town for strolling, eating and drinking.
I have heard from a number of people that Barcelona is excellent as it is a city but with beaches and colourful history and art, relatively inexpensive
Budapest is often hot in September and it has a fascinating culture: we loved just strolling around, going in the ruin bars, wine tasting in the castle and the food was gorgeous. There is a hot spa which is lovely
Extremely cheap for food and drink

As its just the two of you, free from trying to find somewhere to eat that accommodates yours and the kids tastes, I wouldn't dream of going all inclusive! Your free to get out and enjoy a variety of dining in interesting locations, why tie yourself to a hotel canteen. (plus if it turns out to be poor food and you need to eat out lots anyway you pay twice over)

I would look for a destination first that interests you (culture, places of interest, things to do etc) then start checking Tripadvisor accommodation reviews for that area, it can sort them by rating so start checking ones out from the top to the mid level for a consistently good ratings history.

I tend to take more notice of decently written reviews with a balance of opinions. I disregard semi illiterate ones that usually rant about the staff not showing them respect at check-in as they are usually the sort that turn up somewhere and rub the staff (and everyone else) up the wrong way within seconds. Just be grateful if you do go there they probably won't be again!

Then start Googling the hotel, most places will link to various travel agents handling them you can then start to put it all together.

Travel agents can be a useful mine of good information, but they have their businesses interest as their priority so you can't be sure they won't push you slow moving (ie unpopular) destinations or ones with the best margin for the agent rather than the best for you.

At that time of year, anywhere in the Mediterranean will be good for weather. I wouldn't personally bother with the Canaries in the summer months. Bear in mind that because of the troubles in Egypt, Tunisia and now concerns about Turkey, holidays in Spain particularly are getting booked up quickly and prices are rising. We booked our usual hotel in Ibiza last year for this June, and it has been fully booked for ages, so whatever you decide, you need to get booking. Why not look at Thomson, First Choice and Thomas Cook websites to give yourself some ideas. Spain, including the Balearic Islands, Portugal and the Greek islands would be my recommendations. Italy is lovely, but generally a bit pricier.