New Phone

I just got out of my last contract and I have had an LG Remarq that has served me well for as long as I can remember and I want to upgrade to a smartphone. My question is what smartphone would you suggest I am looking for a phone that I am able to take to the gym and have no worry that it could crack or break if I drop it, also it has to have good battery life, and the screen size doesn't have to be that big like the Note an average sized one . I appreciate any advice that you could give me and thank you in advanced. 

Any smart phone with a decent case available. None of them are particularly tough or durable and the battery life of a phone is really more dependent on use case. Yes some phones have larger/better batteries but if it's powering more power hungry hardware, and it likely will be, it isn't going to matter much. So you need to ask yourself what type of apps are you going to be using? Are you going to play music/video throughout the day? Are you going to be using the GPS function or wifi a lot? And are you going to be able to turn it off at any point. Additionally modern batteries are designed for partial discharges and short charges so opportunistic charging throughout the day is most likely the mild inconvenience you'll have to live with if you're a heavy user.

I'd say go into a store and get some play time on some different phones and don't overlook the windows phones, I picked up a Lumia 928 last summer and I'm still very happy with it.

I have the Motorola Moto G and I´m very happy, there is a new version of the phone coming out soon so I recommend it

Similar to Obstinated's comment, I would totally recommend the Moto X. It's basically just 1 step up from the Moto G and has great battery life, tons of awesome features, and unlike most modern phones, is made out a softer, more durable plastic that absorbs more impact when dropped (experienced first hand). The next gen Motorola phones are literally just now being released so the 1st gen X will probably be going for super cheap, or you could just get the next one which I plan on doing and I'm sure is just as good.