Do ira first (roth or standard).
If you are contributing more than the ira max (hopefully!) then try to max the work 401k too.
Be extra careful with that - you said "roth or standard" but if you
or your spouse benefits from any kind of tax advantaged retirement plan during the tax year, you cannot deduct any portion of contributions to a traditional IRA.
source Contributing to a Roth IRA is safe, since you cannot deduct the contributions to a Roth IRA in the first place, it is the distributions which are non-taxable after age 59.5.
Personally I am a fan of 401k's even if there's no employer matching because it comes out of your paycheck before the money even hits your bank account. This is psychologically easier to manage for personal budgeting because you get in the habit of not even thinking of getting that money paid to you at all. With an IRA, you have to earn the money, then turn around and pay it to someone, which sometimes creates a barrier to actually contributing, especially if you have a spouse with other plans for the money.
I am totally OK with the suggestion to do a third in each type of account for FIRE though. After all, the 401k and Roth are of greatest value if you're age 59.5, but they do you almost no good for early retirement, as in they don't help you accomplish FIRE any sooner than that age. So you could do 1/3 in each account type until reaching the $5500 limit for your Roth IRA, at which point you've saved $11,000 in tax advantaged accounts and $5500 in a taxable account. Then perhaps split it 50/50 until the 401k is at the $17,500 max? At that point you'd have put away $17,500 in the 401k, $5500 in the Roth IRA and $17,500 in the taxable account ($40,500 total).
If you're saving over $40,500/year, that's awesome! Tax-advantaged accounts get really tough at that point if you want to avoid crappy government bonds with low yields, so I guess all further savings could go towards some combination of additional mortgage paydown as well as taxable savings. I just hesitate to get too carried away with mortgage paydowns when so many people have 3-4% mortgages and the markets could be doing 4-7%. I guess having too much money and not knowing what to do with it all is a good problem to have though.
Thoughts? Input? I'm personally also working through a new financial situation where I (finally) have enough money to potentially max out a 401k and Roth IRA every year while still having some left over for taxable investing. I don't want to put too much money in one basket or the other though, like over-saving for age 59.5 preventing me from having enough money to retire at age 47 due to insufficient taxable savings.