I love the lines on the Fiat, I don't know how I skipped that in my first comment.
The 850 spider is a really cool little car too, smaller than the 124 spider, but still neat. They're mid-engine, so they aren't as easy as the 124 to work on, but shouldn't be much harder than a 914? basically same mounting place. I haven't driven one, but they're suppose to be fun to drive, they have tiny engines though, 843cc. Not a whole lot of power stock.

As another Petrolicious fan from the UK, I also have to go with the MGB although I would go with the GT version with its Pininfarina designed coupe body. Cheap to buy, massive support network and very tuneable.

I'm not sure if they classify as vintage yet but the front engined Porsches can be brought for that money (here anyway). The 944 would be my pick especially the Turbo.

Are the 944's easy to work on at all? It is always a car that interests me especially when I see them done nicely.

I had a 944S2 with the 3.0 twin cam engine which is a complex unit on account of the need for balancer shafts to smooth out the big 4 cylinder engine. Cam belts and rollers need to be changed regularly, and also watch timing chain tensioner hasn't stretched. Also the gearbox had to be dropped in order to change the clutch.

The driving makes it worthwhile I'm considering getting another although probably look at a Turbo this time.

It's truly much harder now to find value classics but the Porsche 944 definitely is one of these. You can get them in the 6 or 7 grand range. Solid construction, fun to drive. Great, unique sounding engine. These are on the cusp of increasing in value now because of how much the 911 has gone up. Datsun 240 and 280z's are great as well. Not the quality of a Porsche but better styling than the 944 I think.

So many potential answers to this question!! Lots of cool rides for under $7500 - but depends on what you want -- my favorites.
Open top: Austin Healey Sprite Mk II
Muscle Car: Ford Falcon 64ish
Sedan: W111/W108 benz.
Some examples - not mine - no affliation.
http://www.dailyturismo.com/2013/01/5k-flash-1963-austin-healey-sprite-mk-ii.html

http://cgi.ebay.com/ebaymotors/1965-Ford-Falcon-Futura-project-car-GREAT-SHAPE-DONT-MISS-THIS-/160961813116?pt=US_Cars_Trucks&hash=item257a125a7c

http://cgi.ebay.com/ebaymotors/1970-Mercedes-Benz-280-SEL-sedan-/321058453756?_trksid=p4340.m748&item=321058453756&ViewItem=&_trkparms=clkid=5109706238667193522&forcev4exp=true

I'd concur on the open car, spreading it to all Midgets and Sprites 1974 and earlier. A hoot to drive, simple to repair, very few parts made of Unobtanium. The greatest enemy of these are the tin worm and the dreaded previous owners with the concomitant hack job repairs to wiring and other systems.

"Each to his own" - a phrase of humbleness, but also one that would take things into consideration that exceeds personal taste. I'm thinking variations in climate, family situations, local gasoline pricing and so on. Being a devoted petrolhead, it's easy to get carried away in this question and immediately start to search the back of your mind what kind of exotica one could get for this budget. But, if one truly wants to spread the enthusiasm and greatness of this passion to a novice, it might be good to start with something fairly usable, reliable and with decent supply of spares. Something that could lead to a long time devotion rather than a short term fling. For instance, being a fellow northern swede we have limited use for an open top two seater with a crappy heater.. (something that hasn't stopped me though

Well, here's a little something to declare what i have in mind:

For the family:

Early two tone Volvo Amazon. Not exactly unique in my country, but maybe where you live? http://www.curbsideclassic.com/wp-content/uploads/2011/05/amazon-sedan.jpg

Mercedes W114 coupé - stylish and very usable! http://www.google.se/imgres?q=mercedes+w114+coupe&hl=sv&tbo=d&biw=1600&bih=834&tbm=isch&tbnid=xEy-4dMiz-9nYM:&imgrefurl=http://www.autowallpaper.de/Wallpaper/Mercedes/MB_W114_W115/Mercedes-Benz_Strich_Acht.htm&docid=llXfrhRiBPlYYM&imgurl=http://www.autowallpaper.de/Wallpaper/Mercedes/MB_W114_W115/bilder/MB_Coupe_W114.jpg&w=1600&h=1099&ei=LEUCUY6uDKzE4gTlhIG4BA&zoom=1&iact=rc&dur=543&sig=115234709563455344336&page=1&tbnh=140&tbnw=198&start=0&ndsp=35&ved=1t:429,r:4,s:0,i:90&tx=156&ty=31

MG Magnette - http://www.google.se/imgres?q=Mg+magnette&hl=sv&tbo=d&biw=1600&bih=834&tbm=isch&tbnid=5u0kkhYRGA3NYM:&imgrefurl=http://gomotors.net/MG/MG-Magnette-saloon.html&docid=hn7SQDUHAWhhUM&imgurl=http://gomotors.net/pics/MG/mg-magnette-saloon-03.jpg&w=500&h=333&ei=aEYCUeTfFsbj4QTv_YHIBA&zoom=1&iact=hc&vpx=422&vpy=280&dur=1394&hovh=183&hovw=275&tx=121&ty=78&sig=115234709563455344336&page=3&tbnh=139&tbnw=215&start=78&ndsp=44&ved=1t:429,r:94,s:0,i:365

For 1-2 persons:

Fiat 500 (Reliable?? Nah.. but beautiful, easy to work on and with good parts supply!) http://www.google.se/imgres?q=fiat+500+old&hl=sv&tbo=d&biw=1600&bih=834&tbm=isch&tbnid=w7J4IRHT3iF1YM:&imgrefurl=http://lottasplace.blogspot.com/2008_08_01_archive.html&docid=o22JepQ1ydZ29M&imgurl=http://4.bp.blogspot.com/_kVrWB6X9bMc/SKTnTYo4hoI/AAAAAAAAB6Y/6yr9JhqVnBw/s400/fiat%252B500%252Bold.jpg&w=400&h=298&ei=C0gCUf-CEImo4ATA6YHIBQ&zoom=1&iact=hc&vpx=4&vpy=282&dur=2979&hovh=194&hovw=260&tx=142&ty=118&sig=115234709563455344336&page=1&tbnh=143&tbnw=185&start=0&ndsp=37&ved=1t:429,r:8,s:0,i:102

Triumph Spitfire http://www.google.se/imgres?q=triumph+spitfire&hl=sv&tbo=d&biw=1600&bih=834&tbm=isch&tbnid=CXiamegdoqbYWM:&imgrefurl=http://www.klassiker.nu/namnsdagsbilen/grattis-triumph-spitfire-400&docid=bUNW54kNuTEKmM&imgurl=http://www.klassiker.nu/sites/default/files/imagecache/lightbox/oldimages/klassiker/2.-triumph_spitfire69.jpg&w=800&h=385&ei=o0gCUfq5J6qG4ASQv4D4DA&zoom=1&iact=rc&dur=475&sig=115234709563455344336&page=2&tbnh=133&tbnw=256&start=36&ndsp=43&ved=1t:429,r:40,s:0,i:203&tx=197&ty=70

I'm gonna be a little different and say the NA miata, the first year was '89 so it's 24 years old now. Close enough. And for $4k you can get a mint first year Miata.

I think I own (owned) three terrific candidates. My 65 four-carb Corvair Corsa is the most vintage of the group, then my 72 Volvo 1800 ES and now my too new to be vintage, even though it is 25 years old, 87 Alfa Spider. Three very different cars, all of them fun (yes the ES was fun) and all gathered attention wherever they went. That's my 2 cents. Someday I'd like a 71-72 1800 E, and also a Triumph TR-6, but they're not in the under $7500 segment anymore.