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Ok, so I live in East TN, so zone 7a I think. I was planning to start my garden in March, but some stuff happened and it never got started, so now it's May and it's starting to get hot so I'm trying to improvise.

I got five different seeds; arugula, sugar snap peas, mustard greens, Bloomsdale Long Standing spinach, and Evergreen Bunching green onions. Any information on these seeds would be greatly appreciated!

I am VERY new to gardening, and plus I'm doing these all on my 9x4 balcony, so they all have to be in containers. My questions are, when should I plant these, and how big do the containers have to be? I got a few but I think they might be too small...

Well... unless you live up in the mountains, your season is moving along, and the selection of seeds you listed are more suited to the March start date you mentioned. You might be still in that mode?

At this point, it's probably getting too warm/soon or hot for what you listed.
I think it would be better to concentrate on planting beans -- bush beans and maybe pole beans if you have a way to set up some kind of string trellis-- instead of snap peas, consider nasturtiums instead of arugula and mustard greens (you can eat the leaves as well as flowers), Swiss chard instead of spinach. I'm guessing your onions will still grow -- have not tried growing them.

I would try growing cucumbers along with the pole beans on a trellis. I think a nylon netting trellis would be easiest to set up if you can put up an upper support of wire or clothesline ...or a bamboo pole... along the ceiling level.

If you get a lot of sun on your balcony, you can also try growing tomatoes, etc. Get some started plants.

5 gallon buckets would be easiest -- you can usually get food grade used buckets from bakeries and delis. If you want to be more decorative than functional, 12-14" pots and larger, or deep (minimum 8-10" depth) windowboxes would be good start.

The containers should be

SIP and Kratky containers would be best for a balcony because they are self contained systems. You don't have to worry about the water running off the balcony. If your balcony has open rails then light and air will get in. If the balcony has a solid wall, you may have issues with both light and air unless you can raise your plants high enough.

Bunching onions will be fine. As long as your day temperatures are not going to exceed 75 for the next month the Asian greens can take more heat than lettuce or arugula.
Beans, cucumber, tomatoes, eggplant are more suited to warm weather, but they are large plants so you have to have the room for them.
If you get the spinach substitutes like amaranth, NZ spinach, or sweet potatoes, they tolerate a lot more heat.

You can keep most of those seeds for next year if you keep them in a cool dry place, (I put mine in zip locs in the refrigerator). They will be good for next year.
In the meantime look into building some SIPs or a Kratky hydroponic container.
https://www.postoilsolutions.org/documents/Earthbox.pdf
https://www.ctahr.hawaii.edu/hawaii/dow ... method.pdf
https://www.amazon.com/Masterblend-Fert ... B01B8DQ30I

Here is what I use on my deck. Might work for you as well.

I like to get the largest containers possible, but consider maximizing the space by using window boxes over the balcony rails and hanging baskets. Try to grow vertically. You can do some of that by putting your smaller pots on a shelf rather than one or two pots on the floor. If the balcony has open rails then you can get more light, otherwise you will have to raise the plants to get light and air. You might find ideas on how to make a vertical tower garden. The premade ones are expensive and not worthwhile unless you have sun all around but you could make a corner tower and only plant the side that faces the light or use artificial light and shade tolerant plants on the back side.

Thank you all for your suggestions.

Some of you are saying I should plant other things instead of the seeds that I have already. It's a great idea but I'm a beginner, and I don't plan on investing too much money on this if I don't know what I'm doing.

Here is what the seed packets say:
Snap Pea: Early Spring/Midsummer
Bloomsdale Long Standing Spinach: Late summer for fall crop
Arugula: Spring plantings most desirable - avoid summer plantings
Tendergreen Spinach: In mild areas plant in fall or winter (not sure what "mild" means)
Evergreen bunching Onions: Midsummer for fall crop

To give you an idea of summer where I live, it's 75 degrees right now, and I think last year we were getting 70 - 100 degrees. Will I have to wait until summer is over to plant any of these?

I guess I could grow sweet potatoes... The main things I want to grow is the spinach and snap peas, because I eat those all the time. I heard that spinach and greens can be grown in hot weather, but instead of staying as leaves, they will bolt. Is that true?

Mild = frost free
Spinach is a cool weather crop. If you want to grow spinach in summer choose a spinach substitute that can handle heat like NZ hot weather spinach or amaranth. If your weather is 75 degrees now you may be able to get one spinach in if the weather won't heat up for a couple of months. Bloomsdale long standing is somewhat heat tolerant but only till about 80 degrees. If your balcony is east facing and you can put the spinach in partial shade during the hottest time of the day, it is possible. Bunching onions are slow to germinate but the heat won't bother it. You may have to bring it in when it gets cold to keep it from dying back. It is a biennial.

Peas like temperatures 45-70 so you may be past your window of opportunity. It can stand some snow.

Cheap ways to get seeds
Plant seeds from tomatoes. Squeeze seeds from a ripe tomato into a sieve and wash the gelatin off them. Plant them in media. Cherry tomatoes are easier to grow. You will need a large pot 5-15 gallons and you will have to cage or stake them.
Peppers- Get a ripe pepper, either hot or sweet. Remove the seeds wash and plant them.
NZ spinach= check out the specialty markets if they sell the spinach. Get it very fresh and you can plant the stems.
Sweet potato= Get a sweet potato. Put it in a container with water on the bottom and wait for it to sprout. Plant the sprouts. Sweet potato leaves can be eaten as a salad or cooked like spinach.
If you want faster bunching onions, save the bottoms from the store. Take off the hard part of the root on the bottom and replant them.

If you want to succeed as a gardener, it is best to learn all you can about the plants you want.
Spinach, arugula, and peas are cool season crops that don't do well when the temperatures go much higher than 75 degrees. To succeed, you should plant them at the right time which would have been when when the temps were about 50 degrees or started indoors. You also have to take into account your growing conditions. Balcony culture can be challenging if it does not face the right direction. You have to choose plants that can grow under the conditions you have since you won't be able to change the conditions.