Currently my employer offers a Roth 403b plan that I have been contributing to. I wanted to know if it makes more sense for me to contribute to the Roth IRA first rather then the Roth 403b? I currently am not receiving a match on my employers plan yet. Really want to know the key differences.
403(b) plans will allow a larger contribution than a Roth IRA, $19,000 vs. $6000 in 2019. The 403(b) plan will probably be significantly more limited in the securities it can hold compared to opening a Roth IRA with a typical discount commercial broker, quite possibly both limited and poorer in terms of expense ratios and performance. If there is no matching funds from your employer at all, I don't really think that there is much reason to pick one over the other. With the 403(b) tied to your employer, the contributions there will be regular and automatically removed from your paycheck -- psychologically, there is some advantage to effectively never having that money in the first place and enforcing a savings rate up front. But from what you've posted, I don't see any large mathematical advantage one way or another.
Generally you want to first max out your employer's match, then max out your IRA, then max out your 403b.
Since your employer isn't matching, you should first make sure you can max out your IRA annually, then shift to putting as much as you can into your 403b.
The reason to favor the IRA over the 403b is that generally you can get much lower costs in a self-directed IRA at places like Vanguard, Betterment, or Wealthfront.