Hello neighbour! We're practically next door except for a little thing called the Pacific Ocean since I'm on the east coast of Australia.
To start with the simplest of your questions, a new interface will make a big difference to your recordings. The mic inputs on laptops really are rubbish, designed mainly for a headset mic to do Skype calls. They're very noisy and the levels coming out of a mixer would be way too high for them. What interface? Other than saying don't buy Soundblaster (they're for gamers, not serious recording) I'm going to have to say "see below" because there are lots of things to think about in the choice.
I've "been there" trying to get good sounding mixes at a live gig with a feed from the board and it rarely sounds very nice. Ignoring the sound card issue, the biggest problem tends to be that the mix you're getting is designed for the live audience, not recording. Unless you're doing a stadium size gig, most of the audience is getting a combination of live sound from the stage and what's coming through the PA mixer. This means that instruments that are loud by themselves are low in the mix and things that need lots of amplification are cranked way up for balance.
In this case, sometimes a small handheld recorder in a "sweet spot" for the live mix can sound better than a feed from the board.
The "proper" way to do what you're asking with a computer is to get a big splitter box and a big multichannel interface and record every source separately for later mixing. However, I'm going to guess the cost of the stuff you'd need to do this is prohibitive, at least for now.
If you're locked into a real time "as live" recording into one or two channels, one way I've done this is to beg the FOH sound guy for a pre-fade aux feed of your own--even better, two but that's being greedy. This lets you use the "aux send" knobs on each mixer channel to do your own custom mix, balanced for what you hear in the pair of nice "closed cup" headphones you're going to need, independent of what's going to the PA. It'll take a bit of time to dial in the mix (and a few beers for the sound guy to apologise for getting in his way) but results can be okay.
....which brings me to why I didn't specify a sound interface for you. To do the Aux send trick you only need a basic one with one or two line inputs. To go whole hog and multitrack the whole band would take something much more sophisticated but, once you develop the skills, yield much better results. You pays your money and all that cliche.
Finally, just a suggestion the the FAQs at the top of this forum give a really good starting point about what's involved in recording. It's worth making a pot of coffee and settling down for a bit of reading.
Hope some of this helps,
Bob