I used to say cleats but found it uncommon for some people, though I had no trouble with soccer shoes. I have always lived in a Spanish-speaking country (Nicaragua) so I find it hard to know why that is. Are cleats only used in some places? Wikipedia also mentions football boots.
The term "cleat" as a kind of shoe was entirely unknown to me until I encountered the word in Tom Lehrer's Masochism Tango . In the UK they are usually "football boots" (possibly replace "football" by a different sport, but not "soccer", which is a synonym for "football" in the UK). The individual protrusions on the base (which I'm guessing are also called "cleats" in the US) are called "studs" or "spikes".
Edit: I actually supposed for a long time that the "cleats" that Lehrer referred to were some bit of SM equipment, like the "lash" elsewhere in the song. It wasn't until years later that I discovered that this is an ordinary word for spiked shoes in the US. I don't believe the word is widely known in this sense in the UK even now.
In the USA they are almost always called "cleats". (Note that there are different types for just about any outdoor sport, including what we call "soccer cleats").
I am starting to hear soccer cleats referred to as "boots" by some of my fellow players. I think they picked this up from watching English League soccer, and like to say it that way to show how serious they are about the sport. I could see that term catching on in a few years.