I have always given my fish distilled water but I recently got a larger tank. Would it be dangerous to give them tap water with water conditioner in it, if I have been giving them distilled water?
As others have mentioned, the quality of water in different places is always different. To determine if your water is suitable for aquarium usage, run a battery of tests with the water, from pH, KH, to chlorine. You should determine what chemicals are present in the water, the water's pH, chlorine levels, and whether the water is hard water (mineral rich) or not.
A word about hard water - hard water is extremely difficult to control in tanks < 50 gallons and most hobbyists won't use it unless they have a massive system. The problem is that you need to use a lot of buffer salts to get the pH to the appropriate levels, and even then it can still bounce around, killing its inhabitants. The more water, the less severe the spikes and dips.
If you do decide to use tap water, first condition the water on the side - say, in a bucket. When you are first starting out, you want to monitor the chemicals, but later on when you have the process down, you just want to monitor the pH. Let the water sit for a while to ensure that it is stable and has held its properties, then do your 10% change. If you have a way to keep the water from evaporating, you should be able to condition larger batches of water at a time and keep them stable until ready for use.
Short answer: No
But you should probably not do this change at once. The fish might need some time to adjust to the 'new' water. And you should never do a 100% water replacement anyway. So for the first water changes I would still use distilled water, but mix it with normal tap water.
And then, over the next days, you can always remove a few litres from your tank and add tap water to it to fill it up again.
How big is your new tank?
Also, even the water conditioner might not even be needed, but that depends a bit on where you live I believe. Apparently in some places (like the US) there is too much chlorine in the water, which is harmful to fish. In those cases it might be needed to add conditioner or dechlorinator to the water. Test your tap water to check it.
Where I live (Belgium), this is not needed.
But also make sure your tank is cycled, and never replace all the water at once. Maximum 50%.
The other answers have it pretty much covered: gradual changes plenty of monitoring, beware of soft water and hard water.
The only thing I’d add is beware of chloramine. The advice I was given is that in some areas (where I live, UK, soft water) they add chloramine to the water because chlorine smells as it evaporates, chloramine doesn’t (because it doesn’t evaporate). You need a conditioner to get rid of chloramine.
Also, if you have soft water in your area (i.e. lots of insoluble rocks like granite in your area), coral gravel is a fine buffer and will keep your pH high enough for goldfish. My pH plummeted in my first year of fishkeeping. The fish living in the (acid) water coped with the changes but new fish didn’t last until I got monitoring and buffering right.