Hi and welcome to the forum
Get a tank that holds at least 100 litres, the bigger the better with fish. Length and width are more important for fish compared to height. So look for a long wide tank rather than a tall skinny one. Having said this, get whatever you can afford.
If you have wooden floors in the house then make sure they can take the weight of an aquarium. 1 litre of water weighs 1 kg, so a 100 litre tank weighs 100kg. a 200 litre tank weighs 200kg.
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One of the simplest and easiest to maintain filters is an undergravel filter. These are plastic plates that sit on the bottom of the aquarium and are covered with 2-4 inches of gravel. The undergravel filter has a couple of uplift tubes (plastic tubes that go from the filter plate and stick up about 1 foot above the plate). These normally have an airstone/ diffuser in them that is attached to some thin plastic hose called airline. And the airline is attached to an air pump. The air pump sits outside the aquarium and pumps air into the aquarium. Air bubbles up the uplift tube and draws water up with it. Aquarium water is drawn into the gravel, under the filter plates, and is pumped up the uplift tubes and returned to the tank. The dirt is trapped in the gravel and you use a gravel cleaner (see following link) to remove the gunk from the gravel when you do a water change.
http://www.about-goldfish.com/aquarium-cleaning.html
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Another type of filter is the Hang On Back (HOB) style of filter. They are a plastic case that hangs on the front, back or side of the aquarium and use a small water pump to draw water into the case and push it through various filter materials like sponges. They are usually small and compact and don't have the same filtration area as an undergravel filter, but they are usually very efficient and you don't require an air pump to run them (due to them having their own water pump). Air pumps can be noisy, especially the cheaper brands, whereas a water pump is much quieter.
HOB filters are pretty easy to clean. You turn the pump off at the power point, take the filter materials out and squeeze them out in a bucket of tank water. Rinse the pump and filter case under tap water. Put filter materials back in the filter and fill it with tank water. Then turn it back on.
My preferred choice for HOB filters are the Aquaclear range. They are simple to use, have good filtration area and are very reliable.
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The next type of filter is an external canister filter. They are a plastic case that sits next to or below the aquarium and is connected to the tank by plastic hoses. Water drains from the tank into the filter case (which is sealed up and air tight so it doesn't leak). A small pump attached to the filter's lid then pumps the filtered water back into the tank. These filters are quiet and do a good job. They usually have a lot of filter materials (it varies between brands) and can quickly filter the water.
You clean them in the same way as a BOH filter. Turn filter off at power point. The hoses going to the tank usually have taps on them that you turn off. You disconnect the hoses below the taps and carry the filter to the sink or somewhere to clean it. You open the filter and remove filter materials and squeeze them out in a bucket of tank water. Then wash the case and motor with tap water. Put the filter materials back in the filter and reassemble it before reconnecting it to the plastic hoses. You turn the taps back on and let the filter fill up, then turn the pump back on.
Good brands are Eheim and Fluval but there are numerous other brands available too. Power filters and HOB filters usually come with 1 sponge, some ceramic or plastic beads, and a bag of carbon (black granules used to absorb chemicals from air or water). I usually get a couple of spare sponges when I get the filter and have 3 sponges in the filter and do not use the ceramic beads or carbon. You can use the ceramic/ plastic beads if you like but carbon is not normally necessary and is added to filters by the manufacturer so you keep buying replacement media.
Regardless of the filter you get, you need to use a gravel cleaner on any tank to remove the gunk that collects in the gravel. The less gunk in the gravel and filter, the cleaner the water will be. The only exception to this is if you have a heavily planted aquarium. Then you gravel clean around the plants but not where the plants are. However, goldfish eat plants so you probably won't have a heavily planted tank
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If you buy an aquarium in a kit form, they usually have everything in one neat package. However, you usually get a smaller aquarium and cheaper components than if you purchase separate basic model items.
eg: a kit might contain a tank, filter, light and heater all packaged into a nice unit. It will cost x amount of dollars.
You could buy a basic rectangular tank with a decent power filter and LED light unit for a similar price but you would get a bigger tank and probably a better filter.
Light units can be bought from a petshop, hardware store, or lighting store. Hardware stores are usually the cheapest place to buy them nowadays and an LED spotlight can be purchase pretty cheaply and hung above the tank on a wall bracket. Lights are used to grow plants and to see fish when it is dark. Fish do not need lots of light to live or feed. Therefore lights are not something you have to get straight away and any form of light can be used for a basic goldfish tank.
If you want to grow live aquarium plants then get a light globe with a 6500K (K is for Kelvin) rating and that will provide a nice colour spectrum for plants. Globes with a 6500K rating are usually called cool white globes, whereas globes with a lower Kelvin rating (say 3000K) are considered warm globes. The cool and warm simply refers to how people see the colour of the light, with cool being a more bright white colour that we see, and warm being more yellow light that we see.
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re: the water turning dark green in your current tank. That is simply algae. Reduce the amount of light getting onto the tank and do regular water changes to help reduce it. Adding live plants will also help by utilising the light. Algae is simply a single celled plant so if there are no live plants in the tank, then algae will grow instead of higher plant forms.
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Before you buy anything go to a few petshops and see what they have in the way of aquariums and filters, and then post the results here so we can nit pic them. Don't buy anything until we have seen what is available
You can also check Ebay, Gumtree & Craig's List for second hand tanks. However, they don't always hold water and may be contaminated with chemicals. So care should be taken if you do look at getting a second hand tank. If you find a second hand tank and it is still set up and running, then that is possibly something to consider. But if the tank is sitting in a back shed covered in dust, buyer beware
