Wouldnt stand around to see shit start getting tossed around, wouldnt go towards the beast, wouldnt use the bridge, wouldnt climb to the top of anything and would get away from large buidlings and such, for starters. Id also make sure not to follow crowds, shit always happens to crowds. The second someone says "shit I think I saw something move" id be in my car, or running as fast as shit towards the opposite direction thats not the bridge with a large gun, cellphone, CB, walkie talkies, flash lights, lighters, knives, compass, etc, basic survival stuff (and yes I do have those things readily available). If it's an island, there has to be a shit ton of boats around, id get in one of those. I would never get on a bidge, that will be the first thing that gets killed. On the way I would make sure to grab an inflatable raft or a canoe or something by a store I passed. If there were no boats, depending on the time/day/weather/etc I would consider swimming, it's been done, and it's not that hard to swim that length if you just swam on your back. Scuba gear would be the best solution.
If using the the water wasnt possible, chances are id stick around the edge waiting for something to happen. If for some other reason the monster was in the water I would resort to going into one way shelter, that way you know nothing will be on the other side waiting and you have control of what gets in. The shelter ofcourse would have to be a very deep bomb shelter that could take a good bomb/nuke, and id just wait it out. If the monster was only just a water monster, then it could easily be dealt with private planes and chopers. Hiding may be cowardice, but it usually always works.
Never take anyone with you (unless it's a black guy (get it?) jkjk), all it will do is slow you down, use up resources and cause unnecessary attention. Dont be afriad to take out people if they get in your way. The border is always your best bet, but shit is situational and things can change. Im sure there are other things I could think of on the spot, but without any stress/pressure, it makes it harder.