There are a range of platforms that you can use these days that don’t require direct contact with a broker. These are technically forms of brokerage, as they are run by brokers, but many people prefer online platforms because they are fast and easy. You can opt for platforms that offer trading on both the ASX as well as some international markets, such as IG Trading, CMC Markets Invest, eToro or Tiger Brokers. Do your research on trading minimums, fees and other terms and conditions.
If you want to cut out the share platform or broker altogether, then there are lots of ways to buy shares via other methods. You can invest via your super fund, which is likely to have at least some exposure to shares, or you can invest via a managed fund. You may also buy shares through taking part in an Initial Public Offering, in which companies list on the stock exchange.