When you're starting out on guitar, this isn't really an issue you need to think about as you're still trying to get to grips with the instrument. You'll probably just want to stick with a regular string guage and any pick that feels comfortable will do.
As you get to the more intermediate stages and you're beginning to settle on a style of playing, (or many styles of playing) this is something worth thinking about. I wish I was given more direction with this myself early on, as It didn't really occur to me that this would be important.
Stringe guage:
As a general rule of thumb, the thicker a string, the stronger and more defined the tone. You will be able to strike it harder without it breaking it, and the lower frequencies will be easier to hear. You will also have an easier time drop tuning the guitar if you play metal, as the string will hold its form much better when you loosen the tension.
One of the main drawbacks is the string will be harder to manipulate when it comes to bends and vibrato, as you will need more strength to shift it. But once you do acheive this strength in the tendons, you will have greater results. Stevie Ray Vaughan, who is one of the greatest guitarists of all time had immensily thick strings on his stratocaster. His big brawny hands allowed him to really attack his guitar and dig in hard with the pick. This gave him a very strong tone where he was able to play with real conviction. This is why you will want thicker strings when playing genres such as rock, metal and blues.
If you are playing something where the guitar needs to be gentler and further back in the mix, you will want thinner or regular sized strings to allow you to glide over them with the pick and essentially brush the strings rather than attack them. This is better for genres like funk. Check out Nile Roger's playing on 'Get Lucky' by Daft Punk or any of the classic Chic songs like 'Le Freak'
Plectrum thickness:
This is an issue that was even less obvious to me until much later on. I settled with playing with a really thick plectrum to really dig into my strings when playing all that hard rock and metal when I was younger, and when I tried to branch out and play gentler music, my pick really wasn't letting me do it with ease. When I started experimenting with different sizes for different styles it all clicked. It's a similar princple as to the string guage. Thick pick for striking hard, thin for gentle brushing. It's flexibility allows it to bend over the string as you draw it up or down, rather than rigidly staying in place and forcing the string to do all the movement. So as you can imagine this kept causing my pick to get stuck between strings and create really harsh picking sounds that were not appropriate for the style.
There is no rule you have to follow with this, you have to just experiment yourself and see what results you get. What feels more comfortable or what gives you the sound you are looking for.
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