B6 quattro track use..

I like where this is going

Quattro 1.8t with 190bhp seems a good base, weight won't be on your side but it's a good starting point for a track car (definitely one of the better a4's to start with).

Haven't been on a track in mine and most likely won't due to only having the one car (and if I had a second one it would probably be something crazy which would be better suited to the track as mine's only a derv but for the most part the same mods apply regardless of engine).

Handling: rear anti roll bar, decent springs and shorter dampers (or coilovers if you prefer).
For reference mine has an RS4 rear anti roll bar, Eibach springs and AP dampers.

Wheels: Lightest and widest wheels possible with slightly wider rubber than stock or same. Avoid spacers if possible and try to get something made in a strong design/material just to be on the safe side.
For reference mine has Team Dynamics Pro Race 1.2's with Michelin Pilot Sport 4 tyres.

Brakes: Might already come with 312mm discs front, upgrading these really depends on the size/type of wheel being used, probably best to just get some drilled and grooved discs, more aggressive pads and braided lines with 5.1 fluid. If your feeling rich you can upgrade to S4/RS4 bits but take into consideration wheel size when doing this.
For reference mine has 2.0T Special Edition (320mm front, 288mm rear) drilled and grooved discs (forgot the brand), Brembo pads (would have been EBC Redstuff but couldn't get them quick enough) and HEL braided brake lines.

Exhaust: Straight pipe turbo back no question (so long as MOT's aren't an issue, if they are maybe sports cat or find someone friendly), only thing to consider is tracks with noise requirements but I doubt a 1.8t will ever make more noise than a track will specify.
For reference mine has a custom turbo back exhaust with no mufflers or anything that could be a hindrance.

Power: I'm sure there are more experienced people here to talk about this engine but a map is a good start, then there's always K04's, forged internals and all sorts (presuming yours doesn't have a k04? I'm not clued up on 1.8t's in the slightest). May need a clutch too but you'll find that out when all has been said and done.
For reference mine has a bigger turbo, map, Sachs SRE clutch and more but not much is the same as 1.8t's.

Weight/Aero: Best to take the back seats out (permanently if they're not needed on a daily basis), make sure the under trays are on as they help with aero and basically take anything out that's not needed. 1.8T's can most likely tolerate a smaller battery than stock too but going too small will affect it as a daily driver.
For reference mine has buckets, no spare, no CD changer, no covers/dressing under the bonnet and no spoiler or anything fancy on the back (or within the car really other than seats and door cards).

Most important thing is that it's been looked after well, best to carry out every check possible before taking it to a track; last thing you want is your wheels overtaking the car or anything of this nature. Oil, filter, cambelt, water pump, coolant, brake fluid, diff oil, etc, if you don't know the status best to swap everything.

It may seem a big car to track but I've seen much bigger and more crazy, doesn't matter what your in really so long as you take it as a bit of fun.

Then there's the cost... if this is being done on a budget it's best to pick and choose from the above parts and do things as you go.

Honestly - think how many track days you'll be doing.... of any of the A4's, you've got the right base car, it WILL need a RS4 RARB (to dial out some of the understeer), but by the time you've spent enough to turn it into a decently chuckable track car... you could have bought a much better proper track toy for a LOT less and not ruined your B6 as a daily driver....

The B6's are great as daily drivers, but even if you put a RS4 RARB and strip 100-odd kg out of it (rear seats, etc), you'll still have a car that's >1300kg and has ~200bhp... and will eat tyres, suspension, brakes, etc... and those are expensive for A4's.... I started looking at options around this (MX-5, Elise, Boxster, etc).... and decided on a MR2 Spyder - they regularly get run as track cars completely stock: on stock brakes, etc.... and even the stock items last several sessions and the consumables are a LOT cheaper....plus they're fairly cheap to buy and even cheaper to insure (nuggets at £1k and decent ones starting at £2k and mine is only £300pa to insure, incl commuting and parked in Zone4 London! Many people have them modified to around 250bhp and still pay <£300pa insurance.... a Boxster was going to set me back £1200pa to insure (excl track work!).

The MR2 Spyder is mid-engined, rear wheel drive, no traction control, Limited Slip Differential and it's only 1000kg and people quite easily strip 100kg+ out of them, but by removing some of the plastics (frunk/storage bins).... although I'm probably going to add that weight back in with extra bracing and a roll cage

#43 on Evo's Greatest Drivers Cars of all time....

http://www.evo.co.uk/features/features/239692/100_greatest_drivers_cars_countdown_5041.html

Clearly, go for a Lotus Elise (or Exige!) if you have more money, but the MR2 Spyder (2000-2007) is a budget Elise.... and it's in good company in terms of drivers' cars and track toys.

Best way to check is measure the size of the rear anti roll bar, RS4 is 22mm, allot of people rave about them but in my experiance if the suspension, dampers, etc are already upgraded they don't make much difference.

Mine was on stock Sports springs and dampers.... big V6 TDI lump in the nose.... and it used to understeer like a pig.... RS4 RARB (no FARB change) and it transformed the car - it was perfectly neutral.... all 4 wheels would break traction steadily under conservative throttle and the back would step out under provocation - sometimes fairly quickly given the torque in a remapped V6 TDI...