who is the fastest character and whats the best loudout

You referring to raid mode or campaign? Campaign everyone runs the same speed...same with raid

Light weight weapon does increase your running speed and I recommend having a machine gun and shotgun with light weight tag depending on the stage. Special tag magnum-Serpent is the best one but Thunder Rapter is anotehr good Magnum to have. If you do not have those. Steady shot magnum with the Piercing part is recommended.
Grenade Launcher part is a much along with Gluttony 5 and Auto Loaded...Cornucopia 3, Damage 6 and full burst. If playing PS4/One/Switch version, STARS2 and BSAA2 parts are HIGHLY recommended. Furthmore, if you have this version, the blue cased magnums are really good and would put Sonic Assist + on it since it will not take away a slot if you have that tag on it!. It adds piercing automatically with the tag so it is the next best tag next to the special tag variants.

Early raid mode stages recommend characters with good grenade masteries like Quint, O'Brien, Norman etc. Late 30s to 40+ levels, characters with Magnum masteries. Mostly Lady Hunk but Rachel and Raymond are good ones too.