I seem to always make up riffs first then try to put lyrics and a melody to it. Is this right? Is there a right? What do you do?
There really is not right way or equation to the song writing process. Sometimes I come up with a riff and then write lyrics to it or vice versa. Try out some different methods or just to whatever comes natural to you.
Could be either, depending which idea I have first.
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I mostly get the riff first and sometimes melody. I never get lyrics first
Metal is like a apple, no-one likes the core.

Lyrics separately

Lyrics--->Melody + Music = Full Song
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Riffs tend to shape songs. I have loads of melodies and lyric snippets flying around all the time, I apply them when I find a riff sort of thing to contain them.
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What ever works for you is the right way.
I think a lot of songwriters work by having a load of riffs or melodies they've kept over time to use them when a song arises that they work in. And they'll write a lot of lyrics and slot them in when it seems appropriate.
Sometimes i come up with a vocal melody and put chords to it, majority of the time i work like...

Riff --> progression ---> vocal melody ----> lyrics.
Progression ----> vocal melody + riff ----> lyrics.

I have a I VI IV V that i really want to use
Always waiting for that bit of inspiration.
There isn't a 'right' way to make music, because music is art. You wouldn't say to somebody 'you're painting that wrong' because it's their goddamn painting, and he can go about making his artwork any way he pleases. The same applies for a musician. I'm not going to go up to a musician who's working at a piano and say 'you're composing in the wrong order.'

All that said, my bass teacher gave me a wonderful idea yesterday that I went home and tried and found extremely useful.
He told me to start with the melody of the song. Then take the melody and harmonize it, in any way you want. It is from the way you harmonize it that you'll come up with the chord progression to follow the melody.

Example from last night: I came up with a very simple progression in D that shifts between the Ionian and Phrygian modes in tonality. It starts on D, ascending to F# via way of E. So I played a D for the first measure, as F# is the third.
The melody goes up and down for the next three measures but ends each measure on F#. I play an F# minor, but I didn't want three measures of the same chord so I bring it to a B minor, of which F# is the 5th.
The next four measures are similar, but after the the first measure of F# minor, the melody ascends again up to G, then A, then B. I didn't want the chords to stay stagnant, so I moved to G for the last two measures of the round, since B is the third of G major.

It's a damn cool way of writing songs, as it not only starts with the most memorable part of almost any song (the melody), but also forces you to think more about the music involved behind what you're doing. It's as much an intellectual exercise as it is a songwriting session.
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I can see you are a John Mayer fan and I respect you. Anyways, if I write a song, it seems easier to me to get a riff or chords and put lyrics on top of that than trying to put a riff or chords to lyrics, unless you are super pro.
first and last - there are no rules or best way to do it, whatever works works.

i tend to work on both the music and lyrics on and off at the same time. what usually happens is i'll be kicking around an idea for a while, with a few lines of lyrics, something on the guitar, etc, then at some point just enough of it falls together- that first bit of the lyrics and guitar that sync together- and from there it gets easier because finally i have an idea what the song sounds like, and the rest of it starts to fall into place.

the best advice i'd give anyone is to be patient and keep open to how it might turn out, and it will all fall together when its ready to.
I always go Lyrics>Riff>Melody

But that's only cause I have over 30 pages of lyrics that I write while I'm at school
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For me it's a total mix.

sometimes it's riff-chords-melody-lyrics.
sometimes it's chords-melody-lyrics
it's also been lyrics+melody, then chords.

There's no one right way. Don't worry about it.

I will say this, however:

I read a lot of the lyrics people post on the lyrics forum here, and they don't read like songs, they read like poetry. So I'd be very careful about "lyrics first." "Hook first" (the catchy melody+lyrics bit) is fine. But lyrics aren't poetry, they don't always make a lot of sense absent melody, so treat lightly in that department. Make sure you're thinking in terms of music when you write lyrics, if youw ant to do lyrics first.