I currently am funding a 403b but it is Roth Basic from my employer. I also have a Roth IRA from Vanguard that only has 1300$. Does the Roth that I put into the 403b count towards that 5500$? And or, does it pay for me to let the Roth IRA sit there since I am contributing all my money towards the Roth basic 403b
There will be no tax implications of doing a ROTH IRA transfer from one custodian (Vanguard) to another (Fidelity). You will simply go to the institution that you are transferring to and fill out the appropriate paperwork or online form. Fidelity (in your case) will initiate the transfer. Depending on how your ROTH IRA is invested, you may want to sell shares of the funds you are invested in if Fidelity charges you a transaction fee for the particular securities.
Transferring money to a new custodian does not impact nor does it count toward the annual $5500 contribution limit.
Roth 401k / 403b and Roth IRA are completely separate.
Roth 401k (or 403b) comes from your salary and through your employer. There is a limit, but it is quite a bit higher, $18.5k.
Roth IRA comes from your own money and is not related to employment. This part is limited to $5500 per year, independant of the other one.
Both limits are increased for catch-up contributions (when you are over 50).
As others mentioned, you can rollover the 401k part to the IRA, but not the other way round (except as an 'undo' of a previous rollover you did in the current year).
You can't move IRA money into a 403(b) plan (unless the money was rolled over into the IRA from a 403(b) plan and the IRA holds no other money except the rollover amount and its subsequent earnings). A 403(b) plan must be funded through a (voluntary) salary or wage reduction agreement with your employer where you tell the employer to pay you a little less and send the difference to your 403(b) account. All of this is true whether you are talking of a Roth IRA and Roth 403(b) or Traditional IRA and Traditional (i.e. nonRoth) 403(b). Whatever money your employer puts into your 403(b) account at your behest (as described above) has no bearing on the $5500 annual limit on IRA contributions; the limit is the same whether you choose to contribute to the 403(b) or not. What can affect matters is your eligibility to contribute to a Roth IRA at all. High income earners are prohibited from contributing to a Roth IRA at all, period. Somewhat lower income earners are also prohibited from contributing to a Roth IRA at all if they are covered by a retirement plan at work, regardless of whether they choose to participate in the retirement plan or not. So what if you have been contributing to a Roth IRA and discover, after the year has ended and you are preparing your tax return, that you are prohibited from contributing to a Roth IRA at all? Well,you have to remove the Roth IRA contribution (and all earnings from the contribution) by Tax Day -- tell the IRA custodian what has happened and why you want to take money out uf your Roth IRA; don't just say "I want to withdraw $1300 (or whatever) from my Roth IRA" without any explanation -- or you can tell the IRA custodian that the Roth IRA contribution should have been a Traditional IRA contribution in the first place. You are always eligible to contribute to a Traditional IRA. Whether you can deduct the Traditional IRA contribution from your taxable income for that year is a different matter.
Whether you want to leave the Roth IRA at $1300, or add to it over the years, or just withdraw it all and just pay the penalties so as to have one fewer account to keep up with is up to you. What you cannot do is transfer that Roth IRA money into your Roth 403(b) account.
While it is permissible to roll a Traditional IRA into a traditional 403(b) (or 401(k) or 457(b)), it is not permissible to roll a Roth IRA into a Designated Roth Account for any of those. See the IRS rollover table for more details on exactly what you can roll into what.
Were you to roll over from a Roth 403(b) into another Roth 403(b) or similar, the answer to your question would be that it does not count against your contribution limits. Rollovers are always separate from the contribution limits.
You certainly could open a new Roth IRA at Fidelity and transfer the money in, if the goal is simply to not have two places to go; that's entirely permissible and again has no tax implications if done correctly. You'd want to choose the direct transfer option, presumably (as opposed to doing a rollover), as that is simpler and isn't limited in how often you can do it. You should be able to even keep the same securities in many cases if you do it that way.
I'm not sure if Fidelity would let you have just a single login for both kinds of accounts (some places might, some might not), but you may prefer this anyway just for simplicity of having one website to learn.