As a new out of state investor, I’m wondering how best to a accept payments if the tenants want to pay with personal check. Should I just open a bank account at a national big bank so they could deposit it at said bank in their state? Mailing checks seems slow and I wonder if they will get to me on time here in California. I will of course offer them the ability to pay online via Cozy or Zelle, Venmo, bank Bill pay, etc. thanks!

Use smaller community banks. They have less issues with a tenant that needs to deposit cash in your account. Then link the bank to your bank so you can transfer money.

Definitely open up a bank account that they can make a direct deposit into, either by dropping the check off at the bank or transferring money from their bank to yours. Do not accept checks in the mail. I use a fairly good-sized bank but it's regional (Southeast). I don't think it would be an issue to use a national bank, but with most banks you can transfer money without too much trouble. Worst case scenario is you write yourself a check from the other bank once a month/quarter/year. I've done it that way too. 

If I were in your situation, I'd do either auto withdrawal from their account or online payment.  I would NOT give them your bank account to make a deposit for the following:

If you end up having to evict for non-payment, if they deposit $50 into your account, when you get to court they will tell the judge you accepted a partial payment and you'll have to start the process over again.

@Gary Nicholson I would recommend looking for a good property management software based on your requirements and have your tenants set up to Auto-pay or allow them to make their payments online. 

Recommend you charge a fee for them to pay via check. Make the fee high enough to discourage them from doing so or to cover your time & resources to verify it clears, etc.

@Gary Nicholson For the units I self-manage, I have TenantCloud set up. I can market the units, build leases, store docs, complete accounting work. The tenant can submit maintenance tickets, see the status, and pay rent there.  I've used Cozy before as well.  Things work pretty seamlessly :)

@Gary Nicholson , I make all my tenants pay through my property management software ACH transfers are free. I do require first month's rent and deposit to be wire transfer, or cashier's check if I'm local.

@Whitney Hutten thanks!

@Kyle Varga thanks!

@Drew Sygit thanks!

@Mary Smith thanks!!

@Marc Winter thanks!

@John Underwood thank you!