All VFR ATC services are on a time- and workload-permitting basis. It is rare to ever have VFR flight following denied, but it does happen occasionally, usually in the busier terminal areas.
One thing to remember with flight following is that you aren't in the national ATC system computers that are used for IFR flights. The handoffs take more coordination. Again, handoffs are workload permitting, but the controller will tell you what you need to do.
-N1234C, radar service terminated, squawk 1200. For further flight following, try Atlanta Center on 132.62. [This means there has been no hand-off. You can call Atlanta Center and request VFR flight following like you did the first time]
-N1234C, contact Fort Wayne Approach on 127.6. Advise them of your type aircraft and destination. [When you check in, say "Fort Wayne Approach, N1234C, a C-172, VFR KSBN 4,500." They should then come back with something like "N1234C, Fort Wayne Approach, radar contact. Fort Wayne altimeter 29.84. Advise prior to any altitude changes." This is the type of handoff I have heard most often.
-N1234C, contact Podunk Approach on 118.95. They have your information and are waiting for your call. [You're in luck - it's a slow day in ATC-land. All you have to do is check in with "Podunk Approach, N1234C VFR 4,500."]
As for weather information, again, it is all workload permitting. They can certainly help steer you around weather, but don't expect them to look up the latest METAR at your destination.
EDIT - Thanks for the input, Kevin! I didn't know that we had any controllers amongst us!