Fidelity, definitely Fidelity (lively would be an ok second choice).
You can make as many trustee-to-trustee transfers as you want with no penalty. This just means you call one provider or the other (starting with the company you're transferring to is usually advised as they have a greater incentive to get things done), and fill out whatever forms they want you to fill out and they transfer the funds. Sometimes (usually?) the provider you're transferring from will charge a fee for this, so be aware of that if you go this route. You may be able to transfer the investments in-kind. You'll have to check with the provider you're transferring to whether they can do such a transfer from the provider you're transferring from, whether the investments you hold can be held at the provider you're transferring to, and whether there will be any fees for selling that investment at the provider you're transferring to.
The other option is a rollover. This is a process where you withdraw from the old provider and within 60 days deposit in the new provider. You can do this at most once per rolling 12 month period (note that this is not once per calendar year -- you have to wait at least 12 months between rollovers). If you don't then your rollover amount will not be considered eligible for deposit and it will therefore be a non-qualified withdrawal on which you'll pay tax and penalty, and if you already deposited it will have to be removed from the HSA or you'll continue to accrue penalties.
Rollovers aren't hard as long as you're careful about the 12 month rule, and the 60-day rule. First, withdraw from the old provider just like you would for a medical reimbursement. They report all distributions the same, so it doesn't matter what they now. Second, make sure your contribution to the new provider is coded as a rollover contribution. This is the only thing I don't like about Fidelity -- they require that you fill out a form a mail it in with a contribution check to do this. If you contribute online this will be coded as a normal contribution, so the rollover won't be qualified. Third, you'll fill out form 8889 with your taxes indicating the distribution on line 14a, the rollover on line 14b, and reimbursed medical expenses on line 15. As long as 14b and 15 add up to 14a you're good.
The advantage of a rollover is you don't pay any fees and you don't have to figure out the transfer process and get the two providers to work together. The disadvantage is that you can only do it once per year, there might be a bit more work (although transfers can be a lot of work too depending on how well the providers work together), and you definitely won't be able to transfer in-kind.
We let my wife's employer contribute directly to their chosen HSA provider so she doesn't pay FICA on those contributions then do a rollover once a year to Fidelity where we invest. I find the rollover more straightforward since it's just a withdrawal and then a contribution with a form and I fill out form 8889 for medical expense reimbursement anyway (more straightforward than saving receipts long term, and we have more than enough tax advantaged space). The workplace provider also charges $25 to transfer out, so I save that too.
