Here's what I did (and partially recommend) - I started out with a cheap multi effects pedal. I actually got this before I got an amp, so that I could practice with headphones while saving for an amp. It gave me a taste of lots of different sounds and effects, and helped me figure out which effects I liked more and which I liked less. Even though my taste has changed considerably over the years since I got it - I still remember the different sounds it produced.
Once I had an amp with a clean channel and distortion channel that I liked - I got a wah (specifically, a Cry Baby). At the time I barely used the clean channel, listened mostly to metal and just wanted to add a wah for various solos. It was a nice upgrade to my sound.
Later I bought a distortion pedal (Rocktron - Silver Dragon). I half regret this, as I was kinda talked into it by my teacher at the time (he said it would be the perfect compliment to my 7-string I got at the time). I had a lot of trouble dialing in a tone that I liked - and always preferred the distortion channel on my amp.
Fast forward a few years (I didn't play as much for a long time, until fairly recently). After re-igniting my passion for music, I got a bad case of GAS (Gear Acquisition Syndrome). I started building a pedal board, and recently purchased a new guitar.
First additions to my pedal board (besides the aforementioned wah and distortion) were a chorus/vibrato/flanger pedal (TC Electronic - The Dreamscape) and a looper (TC Electronic - Ditto). Both are lots of fun.
After buying the new guitar - I quickly realized how much my old distortion pedal doesn't do it justice. So I bought the Friedman BE-OD Deluxe, which is an amazing distortion pedal. While it sounds so much better than my previous pedal - it makes me see the limitations of my amp (a solid state/transistor amp), since the pedal sounded so much better through the clean channel of a tube/valve amp in the store.
Now my sights are set on the Line 6 HX Stomp - a miniaturized version of the Helix, with both effects and amp modeling capabilities (and it can provide a line out, headphone connection and a USB interface to the computer). This seems like the ultimate upgrade to any pedal board, and a good "toe dip" into the world of amp modelers.
Looking back at all this - it might have been better/more efficient to just start with something like the HX Stomp (or even the full blown Helix) instead of assembling a pedal board (but I am enjoying this journey).
Bottom line - maybe try out a cheap multi effects pedal to see what you like. If you already know for sure that you want a lot of effects and you can afford it - maybe go for a more expensive multi effect like the HX Stomp.