How should I plan for big purchases such as cars and homes while I am still fairly young?

Save, save then save some more!

Easiest way would to be identify the target age/date of purchase and purchase amount+estimated inflation and work backwards. Divide the amount by years/months.

If the amount to save looks huge, then you would have to start looking at how to grow your money or downsize on the purchase or give up on it totally (bummer).

Since you say you are fairly young,

1. Target to grow your earning power/income year on year as much as you can.

2. Save and invest with proper financial planning. Compounding your returns over a longer period. Liquidate investments as and when you need to fund the purchases.

3. Have patience. Continue to maintain discipline and stick to the plan.

Earn More X Spend Less X Invest Wisely

Earn More: Given that you are fairly young, look to maximise how much you can earn, given your skills/experiences.

That said, you can always give tuition in your free-time (it was quite lucrative in my time, just that it definitely ate into my social life), or other side gigs (like even dog-walking).

Spend Less: Cut back on non-essentials. Set aside a target amount you want to save (in this case $34k or more), and work back on how much you need to earn Less spend each month.

Invest Wisely: Once you've amassed enough to pay for your own monthly expense, you can start looking at allocating some of your capital into investments. Many resources online that are free, start small and be informed.