Here is some basic weight reduction that can be applied to any vehicle:
- empty all of that junk/change/food/dirt out. It'll look better too!
- Replace your spare/tools/jack with lighter stuff or take it out. If its a weekend car this is more feasable obviously. You can get a 2800kg aluminum jack out of junkyard Gen 2 RX7's. A puncture repair kit has saved me more than once, that or a can of fix-a-flat could be good insurance.
- If your car weighs less than 2300lbs you could likely take off your power steering an not notice much of a difference.
- Seats are super expensive when you have to buy brackets and harnesses too but they almost always save a lot of weight, or at least move it lower down in the chassis.
- Exhaust. Due to cost of manufacturing and materials OEMs generally don't use stainless steel for exhausts. You can and that can save a lot of weight.
- A/C can obviously be a weight savings but good luck ever selling the car and have fun in August. I have yet to take an A/C system off as much as I would love to.
- Stereo. OEM stereos can be very lightweight or very heavy. If you do a basic system you may come out ahead .
- Sunroof/mooroof. Sometimes you can find cf replacements for the panels and some people simply weld a plate of some kind over the hole. The track, motor, and panel are usually very heavy and the weight is in the worst place on the car.
- Go on a diet tubby!

Next the non basic stuff. These things have a bigger impact on the vehicles dynamics because of rotating inertia and un-sprung weight, this is really the best place to focus on weight reduction because of the dual bonus of static and non-static reduction.
- The flywheel. Probably the best modification you can do to a manual car for the money. You can shave virtually hundreds of lbs off of your car in first and second gear with a lightweight flywheel.
- Wheels. Lightening these creates similar results as the flywheel but to a lesser extent. They are also of utmost importance in how your suspension reacts to bumps. The lighter the wheel(and tire) the better job your suspension can do to keep your tire stuck to the ground over bumps. The static weight removed is not to be ignored either.
- Brakes. Almost all manufacturers use cast iron brake calipers so there is a lot of weight to be lost here if you have the money. Brembo, Alcon, AP, Wilwood, and Stoptech all make lightweight aluminum racing calipers. Try to find one with a dust seal for street use and the most recent design you can afford. Two piece rotors can save a lot of weight as well. Make sure they are high quality floating rotors or you'll be buying them often!
Vehicle specific info:
Miata
On the Miata weight reduction is somewhat interesting. Its already lighter than 99.9% of whats on the road. There isn't much to take off after P/S and A/C so replacement is again a good route. Lighter brakes, wheels, seats, exhaust, and suspension. When I got the Boo-berry it had about 5$ in change randomly strewn around the interior along with some rubber floor mats and a lot of dirt and rocks, so taking that out was a good start! The hood is already aluminum so don't bother with that. CF trunk and fenders would help a bit. If you have a garage and a hard top, you can remove the soft top permanently.
A4
Aside from the basic automotive systems, luxury = heavy, so the A4 is a little more difficult to address than a weekend sports car. If you take out weight you're probably taking out the niceties too. Replacing existing parts with lighter ones is again your best bet. Especially the seats, exhaust, brakes, and wheels which are all quite heavy. The early B5's are some of the lightest modern Audis until the B8 chassis. Everything else is just about hopeless. One significant upgrade is the switch to an aluminum knuckle on the later A4/S4's. You have to change control arms though so do your homework!
The B6 already has some ligher items stock, such as the aluminum front knuckles etc, but the chassis is stiffer and thus heavier. The same route is available to this chassis, CF body pannels, fixed aluminum brakes, two piece brake rotors. Light weight forged wheels.
Accord
On the Accord the first weight reduction I did was replacing the ridiculously heavy stock exhaust with an ebay header, Magnaflow cat, and ebay / Thermal Research &D catback muffler combination. That was done during the engine swap and was later replaced by a Magnaflow exhaust of roughly the same weight.
The next bit of weight reduction on the Accord was the hood. I found a sub-standard carbon fiber hood on ebay for like 100$ one night and I was sold. You get what you pay for as usual. The fitment was poor and I had to get hood pins and some lag bolts to actually get the thing to stay on >_<'. But it was light.
The last weigh reduction I did on the Accord was my forged 14lb wheels. I miss those. After those were on I had room for bigger lighter brakes. Going from the stock cast iron brakes to forged aluminum calipers and two piece rotors saved some a lot of weight as well.
Jaguar XJ8
Forget it. You're not getting any apreciable weight out of this car. Just get a lisence plate like bttlship or dthstar or something. You still want to get weight out? Hmm, well you could sell it and get a X350 chassis which has an aluminum body. You can take out the spare, switch out the wheels for some forged (Good luck with that!). The exhaust is undoubtably very heavy so you can replace that with stainless. You can retro fit the aluminum calipers and buy some two piece rotors. I wouldn't bother with that stuff though. If you're really going that route pick up one of the completely over the top super charged Jaguars of which there are many.
Gutting:
To take out heavy sound deadening tar that is stuck to the bodies of most cars there is an easy trick. You have to freeze it, hit it with a hammer, then chip it out in big chunks with a chisel(plastic covered) or screw driver(if you don't care about paint). I didn't want to buy dry ice in the nice cozy warm summer so I waited until a nice below-freezing- your ass off night in February and went to town chipping like a madman. In a parking lot. At midnight. Apparently the CB7 Accord has more than 20lbs in the front compartment alone. I took quite a quantity of the stuff out of the trunk also, which I did not get to weigh.

This charts where the sound deadening material is on the CB7. I took out the stuff on the floor and in the trunk.

Don't let yourself go! Keep trim!
Weight reduction is one of the ultimate tools in a tuners repertoire. F1, WRC, and all of the major racing bodies capitalize extensively on this. Colin Chapman, the founder of Lotus, used this as the foundation of his automobiles much to the chagrin of other manufacturers.