The very first thing you should do, and something you can do right now, is to set up a savings account, and an automatic transfer into that savings account.
Yes, a savings account will likely only earn you a pittance.
However, having an automatic transfer of money to elsewhere, which you don't touch, will get you into the habit of saving. That alone is a huge step.
You say that your income is about $4100/month, and your expenses are up to about $1500/month. Subtracting $1500 from $4100 leaves $2600 that you could, in principle, invest each and every month. So set up an automated transfer of, say, $2500 per month into that savings account (leaving a bit of a buffer), with a recurrence matching your pay period. Now try to live for at least a few months without even considering that this money exists. If your numbers are accurate, this should not cause any change for you and you should not need to dip into those savings.
For the first few months, don't do more than this in terms of actually investing the money anywhere. You want to get a feel for what life is like living with an actual disposable income of $1600/month rather than $4100/month. You can spend this time looking for and learning more about alternative investments if you want to. This site is a great, free resource to do that; your local library's personal finance section is another.
Since you're currently on a temporary contract that expires in half a year, that's your initial investment horizon. With such a short investment horizon, preservation of principal becomes an overwhelmingly important factor. Once you actually know that your financial future is a bit safer than this, then you can start putting money into other asset classes, such as stocks or bonds.
Mutual funds are not an investment class of their own. Mutual funds is just another way of buying the underlying type of security; for example, stocks or bonds.
Always mind the cost of an investment. For example, look closely at the expense ratio and trading fees. You can't know how any given investment will perform in the future, but you can know what it costs to buy, hold and sell. Actively managed investments are almost always more expensive than a similar investment that simply tracks an appropriate index, while rarely having a better return over any appreciable period.
Figure out a reasonable asset allocation. For long-term investments, a rule of thumb is to subtract your age, in years, from 100, and that's the percentage of stocks you should have in a stocks/bonds portfolio. At 22 years old, you therefore should have about 75-80 percent stocks and 20-25 percent bonds. This isn't an exact science by any means; it's based on long-term past performance, and is meant to give you a reasonable risk-adjusted return, but if you're risk averse, it may be too aggressive, in which case you reduce the percentage of stocks and increase that of bonds (which tend to be a lot more stable, but return less). Split this into a reasonable mix of local and international investments.
Look up what investment accounts are offered by your bank. Look at their fees; whether they are tax advantaged or not; and whether you can freely withdraw money or not. Set one up, poke around, maybe put some money into it to play with (and expect to lose it at first).
Picking individual stocks is hard. Unless you're willing to spend a considerable time tending to your investments, it's probably better to avoid this, and just go with an index fund of some kind instead. If you want to give stock-picking a shot for real, consider having some separate money that you can "play" with.