As a man who has made slow crap cars go fast, I stand against all that has been said so far. Yes, tune the 1000cc!
But keep it sensible and all will be well.
I don't know how much you know about cams but this is basically a simplified rundown. Standard cams are designed for economy and smoothness at the lower end of the rev range. You can get a selection of fast road cams that can be used with no modifications to the head (or the rest of the engine) that increase the top end power but at the expense of a bit of bottom end power. Above that you get cams that develop significantly more power but need work done to head to realise that power, like longer valve springs, bigger valves and bigger reshaped ports. You can really wake up an engine with a fast road cam and it's a great budget modification that you can do yourself. As soon as you move into grinds that need head work, you start to significantly increase the cost and, if it's an engine that already has a good head, it doesn't really need it unless you absolutely want those horsies.
I'm guessing this is an injected engine. ECU remapping is similar to carb re-jetting in as much as you often have to change the fuelling if you change the cam. You can also reprogram the timing on the ECU for more advance, another thing that faster cams like. I do all of this with a screwdriver and a twist of the dizzy, so you ought to consider the ECU remap with the cam.
That's the engine sorted out, now you want to think about making the car go round corners. Assuming you're going to lower it a bit, harder springs, shocks and wider tyres. Make sure the offset is the same, otherwise you'll introduce twitchiness to the steering. Forget strut braces for now. You need them if you're using seriously hard springs on a track which will make the car twist but, unless your car is made out of cardboard, it's really unlikely to twist so much on the road that it merits splashing out on strut braces.
If you just lower it and put wider wheels and tyres on, it'll grip more but its handling will be basically the same. You need to look at its current handling and decide what it currently does and what you want to change. If it currently understeers and you want less, then harden the back more than the front. Conversely, if it oversteers and you want less, harden the front more than the front. You'll just have to make a judgement call on how much, but I uprated my rear springs by about 40% and it all but eliminated the understeer I'd introduced by running a rock hard front end and standard rear. So if you go lower, you'll probably want 25-30% stiffer springs and shocks, then add another 10% to the end you want to sort out.