The national gallery at the bottom of the mound is free to get in and has some really good stuff (when the kids were small we used to do the whole "count the dogs" type thing).
The national portrait gallery on Queen's Street is free in, and usually has interesting stuff on ("when we were young" is great and is on until 13th May). The cafe there is really good, too. mmmm - cake.
The National Museum is really good - you can kill hours in there for no cash at all.
If you're into the gory, the museums at Surgeon's Hall are worth a look - not for the faint hearted. It's about £7 per adult,, £4 per child. iF you happen to be around on 26th May the anatomy museum (in the university medical school on teviot place) is worth a look. It's usually closed to visitors, but it's open that day. The other uni things are also worth a look (the music museum is excellent).
Any day apart from Sunday the Central Children's Library (on George IV Bridge) is open. There's usually something on in there and it's handy if you need somewhere to hang out that isn't trying to sell you something.
You could try this walk - start at Princes Street East End (opposite the Balmoral), then walk up Calton Hill to get a great view over Edinburgh and across to Fife. The kids can climb up on the Athens-y stuff - you can too, I can't because I'm old, fat and have reasonable expectations about what'll happen when I try to get down. Then walk down onto London Road, through to Elm Row - go and browse in Valvona & Crolla, pick up some nibbly bits (or have lunch), then head down Leith Walk. There are loads of interesting shops and places to eat. Really good Pasteis de Nata in the portuguese cafe on the left hand side. This isn't twee Edinburgh though. When you get to the foot of the walk go down Constitution Street and you're in Leith proper. Don't head for the Britannia (it's dear), but you could join up with the water of leith path and then walk up that for about an hour to get you back into Stockbridge, which is another nice place for a walk about, and then walk up through the New Town back to where you started. You'll see some interesting architecture, bits of Edinburgh that visitors don't generally get to, and if its all too much you can just get a bus back into town from anywhere (apart from the water of leith path). If the whole thing is getting tiring by the time you get to Leith there are loads of places to stop to eat and drink (mimi's bakehouse is popular with kids. I prefer the Malt & Hops but kids aren't allowed inside because they don't do food).
Going up the Scott Monument on Princes Street does cost but isn't dear and has a great view. I've been told this by my kids (see old and fat comment above)
My kids love love love the Camera Obscura - it's dear but they think it's worth it even third time around.
If you're around over a weekend try The Filmhouse for kids' screenings if the weather is rubbish - they're less expensive than the other cinemas.
If the weather is good (it's Edinburgh in May, what could possibly go wrong????) - get a number 10 bus from Princes Street (on the same side as the Castle) to Torphin. Get off at the terminus and there's a path ahead of you which takes you up to walk in the Pentlands. There are loads of paths to follow, great views over Edinburgh and Fife. You can walk up onto the Hills and then head East (well, left, but I think it's East) and come back down from the hills into Bonaly and then catch the bus back into town. In effect you end up back where you started, but about a mile further down the road! You can find maps and stuff here: www.edinburgh.gov.uk/pentlandhills/info/15/routes_and_trails
Or you can get the bus (44) out to Balerno and then walk along the water of leith path back into town (it's quite a long way though!). Stuff about it here: www.waterofleith.org.uk/walkway/
You could also take the 41 bus out to Cramond and then walk out to Cramond Island. You'll need to check the tides first, but it's a nice walk and there's interesting WW2 stuff to look at. Be careful with the tides becuase if you miss it you'll end up stuck there.
Buses in Edinburgh don't give change. They do a family day saver ticket that gives 2 adults and up to 3 kids unlimited travel for £8.50. Single's are £1.70 per adult journey; 80p per kid journey and an extra iunlimited adult day ticket would be £4 per day (£2 per kid).
Hope that helps a wee bit.
Have a nice break.