As far as sight reading, let me give you a process for practicing that helps my students learn any piece, and in general teaches you a process for learning music that strengthens your sight reading skills.
1) Play right hand alone and name out loud every note and chord and rest as you play it (if you have chords, rather than saying "A-E-A" or whatever and naming every note in the chord, you can just say chord).
2) Play left hand alone and name out loud every note and chord.
3) Play right hand alone and count the rhythm (this can also help with your rhythm and tempo skills, but you'll just be counting beats--complete with "and"s on half-beats etc.--and assuming you'll be slow for as long as you need to be slow, and you'll be forced to go slower as you figure out how to say something accurate along with your playing, and that's the secret to eventually stay at a quicker tempo, is to stay slow for a long time first).
4) Play left hand alone and count the rhythm.
5) Play hands together and if there are words, sing! If there are no words, you may find yourself still counting the rhythm as you go a lot.
For other instruments: 1) Play through a piece and make sure you know each note. 2) Play through and make sure you know what the rhythm is. 3) Play through as though you're performing and imagine that you will just do your best for today. No matter what else you practice on a given day, if you give at least three steps time and preference, you'll learn.
Other tips: *start at a slow tempo on a metronome and play a whole piece at that slow tempo, which will be very boring but do it anyway. Then move up a couple ticks faster and play it all again. Repeat this until you are at the tempo you want or even a few ticks faster...and then do it all again tomorrow, starting at the slowest tempo and working up. *review "for fun" lots of your old music, at all skill levels, often; playing music your hands and eyes are very familiar with is how you really become good at sight reading. *Learn to play all the major and minor scales on your instrument and arpeggios. That makes you more familiar with sounds your ears are hearing--I like to learn all scales without music but if you also use music and read the scales sometimes, that will also help a little--it helps your sight reading.
As far as sightreading, there simply isn't a short cut for getting good at it other than practice and repetition. It's a bit like learning a foreign language in that you just have to use it until it becomes second nature and you eventually become fluent.
I'm not sure what your main instrument is, but mine is guitar and voice, and I actually found it easier to apply sheet music & sightreading while learning piano first and applying it to other instruments afterwards. As far as picking up new instruments goes, it really just depends on you and what you're interested in. The easiest route is to pick up similar instruments. For instance, if you play a stringed instrument, then picking up a different one shouldn't be a major challenge, same with wind instruments and percussion.
When you will hear improvement is dependent upon how much work you put in. That said, I recommend recording yourself and practicing with a metronome. Start slow and concentrate on getting the tempo and timing down. Every few weeks or so, go back and reference your previous practice recordings and if you're putting the work in, you will undoubtedly hear the difference. Hearing your own progress is a great motivator!
First, I played my trumpet every day practicing drills, etudes, and band and orchestra music. This was regular practice incorporating repetition, slowing down the hard parts, etc.
I also practiced sight reading. My teacher gave me a book of pieces that I would open to a page I had never seen before and play through it on sight. I tried to remember that I was sight reading and not "practicing" in the true sense. So it was important that I played right thru the piece. Once I had done that it was OK to go back and very quickly review parts that gave me problems; but I kept in mind that I was not "practicing" the piece, I was sight reading it.
Keep in mind that normally, a music student will only be able to sight read accurately several levels of difficulty below what they can play technically, and that is OK. So you should look for sight reading materials that are less advanced than what you play and practice regularly.
I hope these tips help.