I know the basic difference between a 401k and a Roth IRA that one is pre-tax and another is post-tax. I've also learned that I can contribute towards both a 401k and Roth IRA. I would like to know if the maximum contribution limit of $18,500 applies to the combined retirement contribution or just the 401k?
Secondly, I would like to ask if I'm (as someone working in private sector and under 50) allowed to catch up on the retirement contributions just like people in their 50's can. I've been employed since 2013 but I only started contributing to the 401k very recently. So I wonder if I'm allowed to catch up for the 4-5 years that I missed since I started paying my taxes or so to say started working out of college.
While pretax 401k's and Roth IRAs are popular, there are also Roth 401k's and pretax (traditional) IRAs. Limits are separate for 401k's (up to $19k in 2019) and IRAs (up to $6k), but apply across pretax and Roth for each. Contributions must be made during the current year for 401k's and by the original tax deadline (normally April 15 the following year) for IRAs, regardless of age. So there is no true catch-up. "Catch-up" contributions for those 50 and older just mean a higher limit -- no actual connection to undercontributing in previous years although some may have done so.
@nanoman answered your specific questions about catch up contributions in your retirement plans.
But if your retirement goal requires you to save more than $25k per year plus contribution limit increases, then there is nothing stopping you from investing in other investments. In this way there is no limit to how much you can catch up. You don't get the same tax advantages as in a 401(k) or Roth IRA, but if you invest in tax efficient funds or ETFs you can have the vast majority of your investment gains taxed at the 15% long term capital gains rate which can be much lower than the marginal tax rates of 25% or more that you have from regular income. It's not as good as Roth IRA tax treatment, but it can be very helpful to have in retirement. Pre-tax 401(k) contributions come out in retirement as 100% taxable income and having Roth and after tax investments to draw from can allow minimizing your marginal tax rate.