What type of lens should I be looking for to take portrait photos?
I would like a lens that portrays a blurred background and also captures a wide background while not making the subject look so far away.
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Your two requirements are at odds with each other at the most basic level when choosing a lens: focal length/angle of view.
The wider the angle of view of the lens is, the harder it is to get background blur with the same f-number and subject distance. The narrower the angle of view is, the further you need to back up from your subject to get a wider background in the scene as well. Backing up also increases depth of field and reduces background blur.
As to being able to "... capture a wide background while not making the subject look so far away", the key there is in your compositional decisions and distance to your subject versus distance to your background.
Re: Michael's answer - you can test this for yourself just using the kit lens that came with your camera.
Take a picture of your subject, outdoors on a dull day - say down the length of a street with shops, parked cars etc.
Set your camera to aperture-prefered mode.
Start at 18mm, aperture as wide as it will go without going too bright.
Frame your subject to just a generous head & shoulders, by walking to where you see that, not by zooming.
Set your lens to its longest, presumably 55mm, leave everything else the same.
You subject should stay where they are, you walk away until you can frame them head & shoulders again, just the same as before.
Though your subject is the same size in both pictures, the background blur & how much you can see will change dramatically.
You ought to be able to see that the portrait of your subject looks 'nicer' in the second picture - the first will have 'big nose, little ears' effect as you were too close [this is why selfies always look bad]
The first, however will have much more background & that background will be much crisper. The second will have less background, but be considerably softer.
You can't have both.*
*Well, you kind of can but it costs an extraordinary amount of money.
As a broad, sweeping generalisation, long lenses make better portraits, 50 - 135mm depending on your style & wallet.
Maybe something to look at is the 'next zoom up' from what you have - a 55-200mm - which would let you explore the possibilities without too much cost.