Maybe it is problem with your battery, bad charging, weared etc. In my opinion 4.2.2 versions gives better battery live. No one will say which custom ROM is best - you have to try it itself. As a new user you can use latest CyanogenMod 10.1.3 (Android 4.2.2)
If you want try custom ROMs on your phone you have to install custom recovery (CWM).
1) Check your baseband version of the phone (for instance XWJV) and find appropriate kernel:
. Check 'USB Debugging' in Applications -> Development.
2) Install it via Odin as PDA file: http://forum.xda-developers.com/show....php?t=1102881
(Download Mode = Volume Down+Home+Power)
2*) If there isn't your version you can try installing 3-file JVZ ROM with Re-Partition and PIT file - then use Semaphore JVZ kernel (but be careful with it, i made it with this version and someone other may give you safer way). Download from http://forum.xda-developers.com/show....php?t=1102881
3) After you got your custom recovery and root you should make EFS backup (IMEI) and copy it into PC - you can use SGS Toolbox for Google Play.
4) Download and copy ROM files and GAPPS into your internal SD - (cm-10.1.3-RC1-galaxysmtd.zip) download from http://get.cm/?device=galaxysmtd
. You can install CM9 first, because going directly to 10.x may lead to loosing your IMEI (i didn't lost it
). You can find Gapps for JB4.2 here: http://goo.im/gapps
5) Turn off phone, go to recovery (Volume Up+Home+Power). Up/Down = Volume button, Accept = Home. Use full wipe (Wipe Data/Factory reset, Wipe /cache, Advanced -> Wipe /dalvik cache). Now 'install zip from sdcard -> choose zip from sdcard' and seek for CM .zip file. After installation do same with GAPPS.
6) Reboot system now - all done. First boot takes a while.