Use the bar chord. Many Weezer songs are just chord combos with some minimal guitar over top. Weezer's guitarist may play the entire chord: you can do the same, or simplify the chord by stripping it down to the three notes of a classic bar chord. Bar chords are easy to play in progression by just moving your hand up or down on the fretboard.
Stick to the major key most of the time. Weezer has little time for the minor key or other stranger sounds. Although there are exceptions (the melody of "Jonas" or the progression of "The World Has Turned" from the blue album), the classic Weezer sound relies more on riffs like those in "Knock Down Drag Out" or "Buddy Holly." Some songs, like
"Island in the Sun," have a debatably more minor sound. Listening to entire albums might help you decide what kind of Weezer sound you want to imitate.
Set up single-note melody riffs. Again, the note combinations are straightforward, major-key "mini-solos" (not much lydian or achromatic sound) that pierce through the rhythm guitar's hedge of sound to grab the listener's ears. Play with these until you hear something you like.
Get the guitar sound by using a mostly clean amp with just a little gain or distortion. The Weezer rhythm guitar has power, but it's not heavy metal. Adjust the guitar (or guitars) until you have the capability of laying down a clear melody line over a pounding rhythm riff.
Add background harmonies. Another Weezer trademark is the clever use of double vocals the main vocal track will be accompanied by a lower harmony track, sometimes with a mostly monotone note sequence. Try this out with two singers to see how much you can duplicate the Weezer vocal sound.