So my first advice is to tackle one thing at a time. You have plenty of time -- you don't need to do everything today.
First things first: focus on your current savings before you worry about what to do with all the stuff you saved in the past. This starts with making sure that you are contributing the appropriate amount to your retirement accounts through work. If that isn't currently set up to make automatic contributions out of each paycheck, that is #1.
Then do you plan any ongoing post-tax savings? If so, set up a brokerage account. Don't get caught up in which brokerage house -- they all offer the same basic stuff by different names. What matters is the underlying fees on the funds you are looking at: lower is better, big time. For that reason, most people here advocate for Vanguard (and that is where I also have my money). You can open an account and have them make regular monthly withdrawals from your checking account. [Note: don't worry yet about IRA, Roth, etc. This is just to get you going -- you can move some of the money to a Roth or IRA later on if you want -- the important thing is to get the monthly savings started.]
Once you do that, now is the time to decide if you want to (or can) open a traditional IRA or a Roth. There are income limits on each, and whether you want to open one depends on how much you want to/are able to save and how much you want to put away pre-tax vs. post-tax. Only you can answer this question. If you decide you want to open one of these vehicles, well, hey, you have a Vanguard account set up! You can just call them up and ask them to walk you through setting up a tIRA/Roth! You can fund it from your existing post-tax Vanguard account, or you can have them take that money every month from your checking account.
Now you have your current investment plan set up -- you have your retirement savings going monthly into your 403(b) plan, you have any post-tax savings going into your Vanguard account, and you have any tIRA/Roth savings going into a separate account at Vanguard. Yay! Pause, have a refreshing beverage, celebrate. Breathe.
OK. NOW you can start to think about what to do with all of those old accounts! See, you didn't have to worry about them right away, because they can just sit happiy wherever they are until you get around to it! So here again, tackle one at a time. Figure out what you can roll over into what (e.g., you cannot rollover "regular" savings into an IRA -- and you don't want to roll over an IRA into a "regular" account because then you'll have to pay taxes and penalties!). Again, the companies you are dealing with can help you with this. Your advisor is correct that some 403(b) plans will allow you to roll in money from old retirement accounts, so if you want everything consolidated in one place, that would be the easiest option (that's what I did). OTOH, if your 403(b) plan has high fees and a poor selection of funds, why would you want to put more into it? In that case, you can take those old IRAs and other pre-tax funds and just roll them into your IRA at Vanguard (so, per above, pick up the phone and call, or hop on the computer -- the intimidation factor is 100x greater than the actual difficulty involved in doing this).
Don't worry about the savings account. Anything that is going to give you noticeably higher returns is also going to come with a lot more volatility. The important part here is to just keep as small an amount here as you legitimately need for your emergency funds/recurring expenses, and to throw any other pre-tax savings at your Vanguard account. (FWIW, all of my Vanguard stuff is in the Vanguard Total Stock Market Index -- 401(k), post-tax, etc. But just pick the lowest-cost broad-based index fund available through your provider and forget about it).
Finally: don't worry about changing jobs. If you change, you may have 3 options: (1) leave your retirement funds where they are (which is fine if they allow it and you like the options); (2) roll them into your new employer's 403(b) (which is fine if they allow it and you like the options); or (3) roll them into your own traditional IRA at Vanguard (which is a great choice if (1) and (2) don't work for some reason).
Good luck! You can do this!