Hello, I am also new here, but have been keeping for around 2 years so I might be able to help you out. First off, let me congradulate you for realizing that pet stores really don't give the best advice when it comes to fish, you have learned that lesson quickly. As for what you will need here are a few suggestions:
Tank - a 29 gallon tank will be a good starter size. You will be limited with your fish selection, but you would still have a few good starter fish that you could choose from.
Filter - if you purchase a kit a filter is usually included, but be careful with this, a lot of times the filter that is included in the kit are not the greatest and very often you can purchase just the tank and a better filter for the same price (or maybe just a few dollars more) I personally would recommend the Aqua Clear 30 (or even the next size up if you want over filtration) for a 29 Gallon tank.
Hood - you will need some sort of hood or cover for your tank as many fish jump and could jump out of the tank if not covered. Also a hood will help prevent spills or other little hands in your case from getting into the tank.
Stand - You will need a good sturdy stand to place your tank onto. You could use an existing table but be sure that it can hold the weight. A 29 gallon tank full of water will weigh about 235 lbs, plus add the weight for your substrate, decor, filter etc. So to be safe you would want a structure that can hold around 250 or so pounds.
Heater and themometer - As you know you don't want goldfish, you will definately need to have a heater and a thermometer. Stay clear of the stick on thermometers as they are not really accurate, go with a floating or digital themometer. As for a heater you should be good with a 150 watt or 200 watt heater.
Decor and substrate - You will need some sort of decor (your daughter will probably want the most input here), the type of decor, i.e natural or themed, really doesn't matter, what does matter is you will want decor that offers the fish hiding spaces and swim through spaces. Also, plants (live or artificial) help here with the hiding places. Substrate can vary as well, there is gravel, rocks, sand, plant substrate, it all comes down to choice. Gravel is the most commonly avalilable at pet stores, followed by rock and plant substrate. Sand you can get from pet stores but it is usually much more expensive and all you really need is play sand (silica sand, the stuff they use in sand boxes - just make sure you rinse it really really well). Regardless of your substrate you will need a gravel vac to clean it. This is espically important with gravel as food and other waste can get stuck into the gravel. If you go with sand the dirt tends to stay on top, but you will still need to suck it off the surface. Also with sand you will need to stir it about once a week to prevent toxic air pockets from forming. This is not that difficult of a job, just use your fingers to sift the sand up.
Lighting - If you purchase a hood lighting is usually built in, but if you go for a plan glass lid you will need to purchase a light fixture. Lighting is important for the fish and also if you have live plants. If you go for live plants post in the plants section for the type of lights you will need.
Liquid test kit - This is vital in order to maintain a healthy tank, and also during cycling. Go with liquid test kits, they are more accurate, and although the cost seems higher they are actually cheaper in the long run.
Other bits and bobs - As already mentioned a gravel vac and thermometer are essential. Additionally, you will want to get water conditioner (that removes both chlorine and chloramine at a minimum, there are also brands that remove heavy metals, this is suggested if you have a high amount of metals in your tap water), food that is approiate for your type of fish (of course)and finally an airstone and pump. An airstone and pump is not a necessity, but it can help move the water around the tank and is a good decoration.
Fish - a good starter fish that would work in a 29 gallon tank would be white cloud minnows, guppies, zebra danio, cherry barbs, harlequin rasboras. However it should be mentioned that generally these are schooling fish and need to be kept in groups of around 6. Also, all the fish mentioned above can live togethor so you could have 2 groups of 6 if you choose to have more than one type.
That should be enough to get you started. You have already mentioned fish less cycling, a good choice, you can do it with out ammonia, you could use fish food, or a raw shrimp, but pure ammonia gives you the most control during cycling.
Best of Luck.