The great news is that it's WAY easier than buying a house. There are many ways of doing it but I'm going to describe what at the time of writing is the simplest and lowest cost in my mind.
Step 1. Sign up to a Broker: Create a STAKE account
Stake is an Australian based broker (platform for buying/selling shares) that is both low-cost (only $3 per purchase! I've been used to paying $20 with Commsec...) and IMPORTANTLY FOR ME gives you a dedicated HIN. The other low-cost brokers at the time of writing had a pooled holding whereby technically you did not have the shares in your name. This is extremely unlikely to be a problem, but then again why take the risk for no reason? I also like the Stake platform because it is so easy to use and clear. It isn't overwhelmed by graphs or colours trying to suck you in to "playing" the share market. Signing up is very easy, just go to their website and follow the guide.
Step 2. Get your Stake account details and transfer money to it
Once your account is setup login to it, make sure you are in the Australian ASX section (you will see the Australian flag in the top right corner, if it's the USA flag just click it and select Australia). Click the "money" icon that is in the top right corner of the screen, next to the magnifying glass, and select "deposit". This will bring up the details of the Account Name, BSB and Account Number for you to transfer money to. Record these in a safe place as you'll be using them every time you want to buy shares.
IMPORTANT STEP - check the price of the share you intend to purchase, as the first time you buy it you need to purchase at least $500 worth of it. Given that some shares cost hundreds and even thousands of dollars per share you may need to purchase much more than $500 worth. For example, at the time of writing VAS (the Vanguard Australian Shares index) was around $95 per share. Therefore you would want to transfer enough money to buy more than $500 worth of it at this level plus a buffer plus the $3 fee. One issue with Stake is that withdrawing the money from your account is not easy or quick, so you want to deposit as close to the amount you need as possible. Therefore I'd recommend depositing the amount that it closed at the previous day plus around 2%. This will cover any unexpected major rises in price without leaving too much leftover money sitting in the account doing nothing.
Step 3. Buy your shares in the middle of the day once things have "settled"
The start and end of the trading day is often quite variable and all over the place. I typically buy my shares at some stage in the middle of the day. All I do is a) login, b) make sure that the Australian flag is there and that my money has been deposited into the account, c) click the magnifying glass and type in the name of the share I want to purchase, d) make sure that the "BUY" option is chosen and select "Market Order" as the order type, then e) type in the number of shares I want to purchase making sure that the "Max amount including fees" is less than the value I have available. If that's all good click the "Review Buy" button then purchase the shares. That's it. Done. This should be quick and easy, but you will be VERY nervous your first time.
Step 4. Update your records
You will then get sent a PDF of the order details via email. Save this to a safe place. Have a simple Excel sheet with the purchases you have made in it, with the dates, share and price per share recorded. This is important come tax time, which is actually much easier than you'd think. Once you provide the Tax Office with your HIN it automatically fills in your tax info for you.
Step 5. Ignore all finance media and do not login to Stake again until next time you are ready to buy.
If you have bought a diversified index fund you are now the proud owner of a tiny piece of hundreds or thousands of some of the most successful companies in the world. You do not need to worry about their success, that is the job of the CEO, CFO, line managers etc. etc. that you are employing at each of these companies. If one of these companies goes bust that is fine, your index fund will replace it with another one. The current "value" of the index of these companies is completely irrelevant unless you need to sell it that day, but why would you? So don't read about it, check on it, or think about it. The only time you check on the prices is next time you are ready to purchase more shares. This is not the only way to do it, but it is the way to do it. The more you tinker the less successful you will be.
Other method of ordering I don't like: To buy shares you can also sit at your computer all day typing in "limits" on prices and waste hours or even days if they are not getting picked up (i.e. you offer less than people are willing to sell at and therefore you don't do a "deal"). Don't fall into this trap. You might save a few bucks some times but you will more likely lose money over time as the odds are the longer you muck around the more likely prices will go up. For shares liked the index funds I use the difference between the buy and sell offers are typically very small and not worth trying to arbitrage.