To continue/repeat:
Yes, most codes require any outlets that are associated with wet environments to be fitted with GFI outlets.
Other outlets are only protected if the GFI is correctly wired and those other outlets are down-stream.
GFI's protect by comparing the current in, with the current out. They need NO ground to operate. If the comparison shows an imbalance exceeding 5mA the GFI will trip.
Any capacitance to ground in the circuit powered thru a GFI provides some amount of this ground 'leakage'. The more the capacitance the more this leakage current. Long cords with ground wires in them also provide capacitance to ground. If the resulting set-up is below 5mA of leakage the GFI doesn't trip. If it exceeds 5mA -> TRIP.
"Surge strips" Have MOVs in them (Metal Oxide Varistors). These voltage spike protection devices have capacitance.
Motors also have this capacitive leakage as do lighting ballasts.
Even more confusing is when some devices start up they can cause small brief spikes. This means some people have "flaky systems" that trip their GFIs "randomly". They trip because their systems have a lot of leakage capacitance that is adding up to, maybe, 4.5mA and when their heater kicks on, once in a a while, 5mA is exceeded and the GFI operates
So while it would be better boojumsnark25, to just install a NAME BRAND** GFI as code would dictate, your plug-in version is the next best thing. Keep in mind they are often a little more sensitive than the receptacle versions. They often trip a little sooner than the wall GFIs would.
Plug in your portable GFI then plug in your surge suppressor. No trip? Plug - into your surge suppression strip - your various pieces of equipment one at a time. If any one item trips the GFI unplug everything from the surge suppressor strip and after resetting the GFI try the device in question. If it still trips unplug the surge strip and plug in the offender to the GFI directly. If it still trips and it isn't a device where you can easily look at the wiring (like a light hood) cut the cord off as close as you can to the device and throw it away as it is dangerously faulty.
Otherwise if you get everything plugged in but the last few things when added trip the GFI you need to remove the surge suppression strip as its inherent leakage is consuming too much of your GFI's allowance. If you still want surge suppression move it to in front of the GFI. Get a NON suppression power strip to provide the needed outlets.
One very serious issue with plug-in surge suppression devices that you are contemplating is this: Many of them trip on power loss. This is because of their pedigree. They were originally created for building contractors and construction. Often generators are used or the power is shut off at night. The fear was someone would leave a power tool ON and when power was restored, after any kind of outage, someone would be injured. So a lot of them need to be manually reset after a power loss. What does this mean to you? If you are at work or it's the middle of the night any brief power failure will shut off your system permanently until you manually intervene.
**Name Brand means a Hubbell, or Leviton, or Standard.
DO NOT buy Kingswhatever or "Jiffy Safe" these off-brands, on a safely device, this critical, are a foolish waste of money.
Before I knew this I bought whatever was 'there'. I had one go insane and burn out in a cacophony of deafening buzzing while I was on the phone. This was about a week after I installed it and nothing was even plugged into it.
The big names I mention above have a name to protect and so have test standards and manufacturing quality controls implemented. That's why their units coast a few more dollars ~$4. You get what you pay for.
One other thing, power strips do not always trip.
