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It would be luxury to record in a room where there was no sound coming in from outside, and no sound leaking out to annoy your neighbors. But for most of us it isn't a practical possibility to provide complete soundproofing. However, there are things you can do quite easily that will improve your soundproofing to a significant degree.
Some parts of buildings are easy to soundproof; others are more difficult. It's straightforward to build a wall with good insulating properties out of bricks or dense blocks (but not lightweight blocks, or blocks that are porous). It's not too difficult to build a partition from drywall (plasterboard), providing you use enough layers and seal all gaps well.
But the difficult elements are the doors and windows. The windows are perhaps most important because that's where sound will leak outside of your property. The door might just let sound leak to an adjoining room.
So how do you soundproof a window? The easy answer is to double glaze it. Commercial double glazing is remarkable cheap these days, and it does offer significantly improved insulation over a single pane. If the window you have installed is fixed and not capable of opening, then you will be well pleased for the money spent.
But if the window does open, then that's a problem. (Of course it will have to be closed during recording otherwise you'll have no insulation at all!). Now the degree of insulation is primarily governed by the seals around the window.
Some windows have so-called 'brush' seals, in which fibers inter-mesh when the window is closed. This provides a reasonable seal against drafts, but not against sound.
For soundproofing, you need a 'compression seal'. In this, a rubbery strip is squashed when the window is closed, closing off any direct sound path. Commercial double glazed windows do commonly have compression seals, but they are often not particularly good at thei job.
To test this, apply some force to pull the window tightly closed. You may be able to hear a significant improvement. If this is the case, then you need to beef up the compression seals with thicker and firmer rubber strips.
You know, soundproofing isn't rocket science. You need lots of mass, no gaps and good seals. There's not much more to it than that.