My audio tips (both soundboard/audience recordings)

Bon Jovi tour:

Philadelphia 84 (sb)

Fahrenheit tour

Document in Japan-Tokyo 85 (sb)

Slippery tour:

Cincinatti 86 (could be 87) (sb)

New jersey tour:

Tokyo 88 (sb)
Phoenix 89 (au/sb)
Rotterdam 89 (sb)
Chili 90 (sb)
Rio 90 (sb)
Hammersmith 90 (au)

KTF tour:
Milton keynes 93 (sb)
Rotterdam 93 (2nd night) (au)
Copenhagen 93 (au)

Cross Road tour:
Wembley 3rd night (sb)
Osaka 95 (au)
Tokyo 95 (au)

These days tour:
Tampa 95 (au)
Yokohama 96 (2nd night) (sb)

Crush tour:
Zurich 00 (sb)
berlin 00 (au)

One wild night tour:
Yokohama 01 1st night (au)
Melbourne 01 (sb)
Giants 01 (sb)
Cologne 01 (au)

Bounce tour:
London 02 (sb)
Hyde Park 03 (au)
Yokohama 03 (au)
Tokyo 2nd night (au)

Borgata shows:
TLFR (dvd sb)
Box set show (sb)

HAND-tour:
All of the Japan 06 shows are excellent audience recordings
Atlanta 06 (au)
Amsterdam 05 (sb-but only because it's soundboard)
Atlantic City 06 (au)
Toronto 06 (1st night) au
Philadelphia 06 (2nd night) au

solo gigs

JBJ:
London 97 (sb/au mix)
Paris 97 (sb)

Richie sambora:
San Diego 1992 (sb)
Hamburg 1998 (au)

most of these shows have good to great setlists, athmosphere and/or excellent perfromances.
Imho, these are the can't go wrong shows shows. There are better shows out there setlist-wise (like Yokohama 96 3rd night compared to the 2nd night), but for starters these are highly recommended to get a good overall impression of BJ live in good, very listenable quality