If your looking for a quick fix, the only reliable one is to transpose the songs you want to sing. Even in areas with trained singers such as opera and broadway, the songs were written in keys comfortable for their first performers. Even today, some revivals of broadway shows may adjust particularly rangey songs up or down a small amount. As for anything in the popular realm (rock, pop, etc. anything with guitars…) transposing is easy, and extremely common. I've learned to play some songs with bands in 2 or 3 different keys, depending on the singer.
However if you're willing to give it some time and effort, do some vocalises (like the warmup 5-4-3-2-1 Monica Cellio mentioned above). There are a great number of these, and you can make them up yourself. When working on range, you especially want to make sure you're singing correct; relaxed and in a healthy manner. Too many people try to press and force the voice beyond what is comfortable for it.
You mentioned you're an instrumentalist, but another thing to keep in mind is that trained singers don't regularly sing at the edges of their range. Most singers know their highest and lowest notes, their highest and lowest reliable note, and neither will be regularly used (in most cases). For example, though I'm far from a trained singer, my highest note full voiced is tenor Bb. I can rarely ever get that high though, and it's not pretty. My Ab's are more reliable, but I would not sing that high solo; I'd transpose. G is the highest I'm comfortable singing in a solo song, however if the transposition is easy, I might take it down to F.
If I were putting more time and effort into singing (as I would do if I land a choral teaching job) I'm sure I could strengthen my voice up to the Ab to make everything more reliable, relaxed, and pleasant-sounding, however it's doubtful my overall range would increase by all that much.
Finally, I don't know how old you are, but the human voice doesn't fully mature until the late-20's or even early 30's, depending on your voice type, so don't push anything that's beyond your range right now.