Should I save a percentage of my weekly earnings and put the aside for SARS?
Yes.
What percentage should I allocate?
That depends on what you earn. Ideally you'd calculate the tax due on your extra earnings, and allocate that. As a fast-and-nasty alternative, 40%. The highest tax bracket (for people earning over R673100 per year) is 40% of the amount above R673100, so the highest possible tax you'd have to pay on your Uber income would be 40%, and that only if you already earn R673100 or more annually from your day job.
Will SARS see me as a higher income earner due to my Uber earnings
Yes. Your employer is deducting income tax based on your salary. If you earn only your salary, then that will be your total tax due. If you earn income apart from your salary, SARS will you you as earning the higher income (i.e. the sum of your salary and the additional income).
and push up the tax brackets?
Maybe.
If you earn less than R174550 annually, SARS will take
18% of your income.
If you earn more than R174550 annually, SARS will take
18% of the first R174550
25% of the amount above R174550
If you earn more than R272700 annually, SARS will take
18% of the first R174550
25% of the next R98150 (272700-174550)
30% of the amount above R272700
If you earn more than R377450 annually, SARS will take
18% of the first R174550
25% of the next R98150 (272700-174550)
30% of the next R104750 (377450-272700)
35% of the amount above R377450
If you earn more than R528000 annually, SARS will take
18% of the first R174550
25% of the next R98150 (272700-174550)
30% of the next R104750 (377450-272700)
35% of the next R150550 (528000-377450)
38% of the amount above R528000
If you earn more than R673100 annually, SARS will take
18% of the first R174550
25% of the next R98150 (272700-174550)
30% of the next R104750 (377450-272700)
35% of the next R150550 (528000-377450)
38% of the next R145100 (673100-528000)
40% of the amount above R673100
Your Uber earnings will cause you to move into a higher tax bracket if and only if your total annual income including Uber earnings is more than one of the amounts indicated above, and your total annual income excluding Uber earnings is less than that amount.
Please note that earning more money will never (and can't possibly) increase your tax by more than your increase in income. Earning more before tax will always mean taking home more after tax. Only "how much more" will vary. The higher tax rate is only applied to the amount over the tax bracket threshold, and the portion below the threshold is taxed at the lower rate.