2nd Gear and Warning Lights

First my 2nd gear problem - when I put it into 2nd and if I go over half throttle the gears skip. Doesn't make a very nice noise. But if I just slowley give it gas in 2nd and shift up to 3rd all the other gears work fine. What would be causing this problem?

Warning lights - When I start up the bike it goes through the warning lights and when it does my brake light and battery light stay on. The fluid level for my brakes is good, and the battery is new. With the battery sometimes when I go to start it it just clicks and wont start, but if I roll the bike in first and release the clutch I can put it back in nutral and it will start. So wondering if it would just be connections that I need to look at??

 

Jumping out of 2nd can be a PITA and sounds horrifying when it happens. You may want to check the adjustment of your shift lever - if not correct you may not be getting full engagement after transiting neutral on the way up. If it seems OK you're probably looking at worn shift forks or grooves - not sure if you can address these issues without a complete teardown, hopefully I'm mistaken (its been over 10 years since I had mine torn down that far).
Your brake light issue may just be a bad connection from the master cylinder to the harness - just follow the wires from the mc to where they connect (might be inside or just outside the headlight housing). I think the switch is normally closed - you can check it with a multimeter or circuit tester, or link the two leads from the harness and see if the warning goes away. As for the battery, do you have the battery sensor installed? If not, or if its pooched, there's lots of good info posted for an easy-to-wire battery bypass.
A general harness and connector inspection/clean-up may be in order - I like to pull the connectors apart, clean and inspect the terminals, then reconnect after applying a little dielectric grease where appropriate. My bike was clicking when the start buttom was pushed for a while, but the problem cleared up after cleaning the harness connectors, fuse contacts and fuses, and giving the starter solenoid (the round one mounted just behind the battery) a little shot of WD40. When the problem was occurring, switching the bike off at the key and turning it back on again would give me one operation of the starter - never tried bumping it in gear, then attempting the start buttom - if I put it in gear it was always to 'push start' it.