It depends what your after and how important surround sound is to you. I've had a 3.1 system in the bedroom up until the recent passing of my wife and I can tell you that 3.1 is not surround sound. Its way better than the TV's speakers for sure but its not surround. You need the surround speakers as well.
People argue that the center channel is the most important speaker in the system but there are some qualifications behind that arguement that needs to be addressed in order for that to be true. The center channel is supposed to anchor the main dialog to the display and this is very important if your listening positions are greater than 30 degrees off center from the display. However, if your listening position is narrower than that, you can forgo the center channel and use the "Phantom Center" channel in your AVR's setup to put the dialog across the front left and right main speakers. It depends on how good your left and right main speakers are. I've had to stash my center channel away to stage a house for sale and I never missed it the center channel because my listening positioning was narrow than 30 degrees. It really worked well.
I'm telling you all of this as there are strategies to for you to follow to get you your 5.1 system but not all at once. You will need to budget for a yearish to complete the package
- If your listening position will be under 30 degrees for at least a year, forgo the center channel and pick up the surround channels. Save up and get the center channel the following year or whenever you get the money.
- If you think your listening position will be greater than 30 degrees, then budget for the surrounds and get them once the money comes in.
I hope this helped.
Once last thing, its very important from a timbre matching perspective to ensure that your front 3 speakers come from the same manufacturer and even better if it comes from the same lineup from the manufacturer. Do not mix manufacturers because pans across the 3 speakers will sound disjointed and the AVR room correction facilities cannot fix this.