Having (or contributing to) a 401(k) does not affect the tax benefits of your Roth IRA. They can affect your AGI and your ability to contribute, though (e.g. 401(k) contributions lower your AGI, which may make you eligible to contribute where you were not before).
The contribution limits to both are independent as well - so (if eligible) you can contribute the maximum to both accounts. The limits apply to all IRA accounts and all 401(k) accounts, though, so you could not contribute the max to both a traditional IRA and a Roth IRA in the same year.
That said, you might also consider a 529 plan instead, which also provides tax-free growth. There are pros and cons to both, but some states give state income tax deductions for 529 contributions, and it has higher contribution and income limits. The main drawback is that it has to be used for education expenses, whereas a Roth does not - it's intended for retirement but form some reason education expenses are penalty-free as well. Depending on your state, the investment choices may be limited as well.