Hi there,
Welcome to the fish keeping hobby!
Allow me to provide info a couple basics many beginners
dont know about, so as to avoid a troublesome beginning for you.
When you first get a new tank, when you set it up and add livestock,
the tank will go through what is called the Nitrogen Cycle.
As the fish pee and poop in the tank, ammonia will build up.
Ammonia is lethal to fish. However, soon after ammonia builds up in
the tank a type of bacteria called Nitrosomonas builds up to turn
that harmful Ammonia into less harmful NitrItes. Ammonia starts to disappear
off your test kit and NitrItes rise to near lethal levels. Then, another type of Nitrosomonas bacteria builds and eats the NitrItes, resulting in the end product NitrAtes, a relatively harmless chemical that feeds live plants.
The cycling period takes approximately 21-60 days, depending on how many water changes you do, how often, and how much bioload (the amount of waste and size of fish in the tank) that are present.
Neon Tetras, while beautiful, are fragile and will not survive the cycling process. There are however fish that will.
You should ask your local fish store, if you can return fish in trade, after your tank has cycled. Tell them you would like to purchase 6 zebra danios,
to cycle your tank, and would like to return any living danios after the cycle back to the store for either credit, or trade for Neons.
Zebra Danios are about the toughest, hardiest fish there is. They will usually survive a cycle, without any problems, where many other fish would die.
Now, you need to get test kits for Ammonia, NitrItes and NitrAtes,
or a master test kit that includes these. Dont buy the dip stick ones, they are faulty, dont read right, and need to be replaced monthly as the reagent starts to detiorate upon exposure to air. Stick with the liquid drip tests. They will last you years and be reliable too.
This will help you measure the levels in your tank.
Once your ammonia has disappeared to 0 and your NitrItes have dropped to zero, and NitrAtes have risen, your tank has cycled and is ready for your stock.
During the cycling period, when ammonia reach 2.0 ppm, you will need to do a small water change. You generally should try to do water changes only every 2 weeks while cycling, if you can. But if the ammonia gets high do a 25% (of the tank volume) water change.
Always replace water with Dechlorinated, Same temperature water.
A great Dechlorinator is Prime. Its the cheapest in the long run and superior beyond all others. 1 capful treats 50 gallons of water. It removes Chlorine, Chloramine, heavy metals, and detoxifies Ammonia and NitrItes. It is great to use during the cycle , every few days to keep the ammonia less toxic to help the fish through.
Now, once your tank is cycled, you can add your fish.
Definitely 10g has room for some neons and some other fish.
The general rule of thumb for stocking fish is 1 inch of adult fish,
per gallon of space.
Neon tetras are schooling fish, so should be kept in groups of 5-6 of more.
As neons are about 1 inch at maturity, if you got 5 neons, that would be 5 inches of adult fish, allowing you approx 5-7 inches remaining to work with.
There are many fish that are suitable with neons. I suggest a contrasting tetra or similar, perhaps the nice yellow Lemon Tetra, or the pinky with white gold and red White Cloud mountain minnow, or perhaps the slightly larger cousin of the neon, the Black neon, which is black and sports a sharp white stripe.
You could do two schools of 6 small 1 inch fish each in your tank, if you are willing to do 35% water changes with gravel vaccuum of 50% of the substrate every 7-10 days, or every 10-14 days if you have a fair amount of Live Plants.
If you do not have live plants, stick to the schedule of every 7-10 days.
Never remove too much water or vaccuum 100% of the gravel, as that will harm the nitrosomonas bacteria that live in your gravel, and may cause your tank to go through a mini cycle.
Another option to stocking, could be to go with some neon tetras,
and then choose a centerpeice fish, a small, colorful fish, that can be kept individually, and usually has a fair amount of personality.
A male betta (siamese fighting fish), or a Dwarf Gourami, Or a Paradise Fish,
all come to mind. All are bout 2.5 inches at maturity, peaceful so long as kept with no others of their kind, and do well in a smaller tank. Beckfords/Dwarf Pencilfish are very interesting, and can be kept in pairs or trios. Could definitely be a centerpeice fish. There is also the Blue German and Bolivian Rams, smaller cichlids that can be kept alone in a small tank. The blue is more colorful, but the bolivian more hardy. They grow to 3 and 4 inches respectively.
Or, maybe you like catfish. If so, the Corydoras group of catfishes include many varieties that are smaller, and suitable for a 10g tank.
Cory cats are schoolers as well, so of course you would want to keep a group of 5-6 with your group of 5-6 neons.
Several of these are very large fish, over 3 inches and even 4. Avoid these particular few :
Albino or Bronze
Pepper /Paleatus
Metae/Bandit '
Emerald Green /Brochis Splendens
They are the largest of the cory family and reach 3 or more inches, and need at least 20g of space. The emerald green is not really a cory, and grows to about 6 inches.
Pretty much every other cory catfish is small enough for a group of 5-6 in a 10g with some neons.
Some are particular small, such as the Pygmy Cory or the Hasbrosus Cory,
both coming in at 1.5 inches or less. They are super cute!
Also, there a 2 different types of Loaches, which are kinda like catfish, that would work in a 10g tank. One is the Dwarf Chain Loach, species Botia Sidthimunki, which grow to about 1.5 inches.
The other is the Kuhli loach, an orange and black striped fella that looks almost like an eel. They are about 4 inches long, but very thin, so would count them as only about 2 inches of bioload each. Loaches are Shoaling fish, slightly different that schoolers, preferring some members of their own kind, with 3 individuals being the generally lowest accepted number for keeping together.
There are also many other ideas, like, Guppies, Platies, many tetras, many rasboras, killifish, mosquitofish, etc etc.
Also, there are non-fish critters, such as Shrimp, African Dwarf Frogs,
and even crayfish(but these eat fish so care must be taken that they are not mixed with slow, less intelligent fish like guppies).
Mollies of all species are too large to fit in a 10g tank. Most mollies reach 3-4 inches, and some as much as 6. Mollies are also much more difficult to keep than their cousins the Guppies, Endlers, and Platies.
Also, in case you were rolling the idea of goldfish around anywhere, dont.
No goldfish are small enough to fit in a 10g tank, goldfish require specific setups, with 30G being the minimum for 1 individual.
I hope this is enough to get your started.
Remember, the fish you are choosing are tropical, and will need a heater.
Most will also do best in a neutral PH , between 6.8-7.2 being best.
Filtration is also important, and, if i didnt mention above, never change a filter cartridge when doing a water change, just rinse in used tank water and pop back in. Only replace when absolutely needed, when filter speed is impaired or tank looks dirty.
Also, dont overfeed your fish. The size of their eye is the size of their stomach. Feed accordingly, only what the fish can eat in 3-5 minutes time,
and also siphon off any uneaten food.
Also, try to vary foods a little, say a regular flake, a veggie flake, a shrimp pellet, some frozen bloodworms. This will help your fish grow strong healthy and beautiful.
Please dont hesitate if you have more questions, and heres a link to a description of the Nitrogen Cycle you may find useful.
Good Luck!
http://faq.thekrib.com/begin-cycling.html
And a link to a pair of actual Fish only Forums, which you also may find useful:
http://www.fishprofiles.com
http://www.petfish.net