You've gotten some good advice already. I wanted to add my $0.02 here though because two of my three cages are simple plastic pans (converted critter nation cages) instead of a grate setup. I have a third cage (Eva's) with a grate and I've also had a lot of foot/grate/cage experience in wild bird rehabilitation and spending awhile working for a pet store. There are some pros and cons to either choice, I'll break it down - based on my opinions and experience - very quickly and simply...
GRATE
Pros: birds can't get to feces, birds can't get to spoiled food, discourages birds from tearing up papers at bottom of cage, in some cages required (if there is a gap when only tray is used)
Cons: grates are messy and harder to clean, birds can't forage dropped food and toys, birds can get sores if they spend too much time walking on grates, birds will have no flat surface option (unless you have flat platform perches)
TRAY ONLY
Pros: birds can forage, get to any dropped toys, easier to clean than grate, birds should not get sores walking on flat surface
Cons: birds will have total exposure to poop and dropped spoiled food, requires more frequent cleaning, birds can get to bottom papers to tear them
I have seen birds - especially doves and pigeons who tend to spend lots of time on the ground foraging - get nasty foot sores from grates. If I was using a cage with a dove and the grate was not removable, I added paper towels or newspaper to help smooth out the surface. In parrots, the same can happen if they spend a lot of time on the grates. A flat tray is a bit better for their feet but it doesn't matter if you can encourage your bird to use platform perches and spend less time on the bottom. I have even used dog frisbees (plastic and durable fabric) to make very affordable platforms for Eva and she loves to roll around and play on them!
If your bird can get to the paper at the bottom of the cage (even if it's behind a grate) make sure any print is with soy-based ink. Most newspapers only use soy-based ink now but never assume. Regardless of whether you have a grate, a bird will be exposed to much more ink chewing and mouthing the paper than walking across it. If your tray is low enough from the grate that the bird cannot reach the papers this is less of a concern.
EDIT: While we're talking about feet I just feel the need to add this for anyone reading. Please always check your bird's feet for human hair. I once was given a pair of ringneck doves who had been (neglected) family pets for years. The hen had a long strand of hair around her foot and she was unable to remove it. Over time it embedded itself into her foot and cut off circulation. By the time we took the pair in, the infection had spread to the rest of her body and while we were able to remove the foot she later died of sepsis. The male was in better condition and ultimately able to find a loving home but I hope her story will serve as a reminder of another thing to be wary of!