Just started to play and reached 15 strength killing chickens, should I continue until I hit 20 strength, 20 defense and 20 attack before moving to cow field?
Got training sword and shield which will be used until I will collect more feathers and upgrade gear.
What type of items should I get once I hit level 20 playing as melee character?

F2P or P2P?

If you're F2P, I'd suggest you go to Port Sarim with your cash stack - buy from the Magic/Rune shop Blue + Black wizard hats, bank them and repeat. Wizard hat = 2 coins in shop, 283 in Grand Exchange.

I'd suggest you go kill cows already, if you've got absolutely no coins, loot the cowhides, walk to Al Kharid, tan and bank it - repeat.

Almost 20 strength, will follow your advice and move to cows.
Cant buy hats for the moment as I have zero gold. Will collect cowhides and post when will get those attack skills at 20.

20 att - 20 str - 20def
Getting range now up to level 20, should i raise magic too after?
Cant use Grand Exchange yet but still good can grab things for mobs and sell later for a bit of gold.

Look up the 2 quests: Romeo & Juliet + Ernest the Chicken. with these 2 quests completed you will be able to use Grand Exchange for runes etc

I'd wait with Magic till you have got 50k+ coins for runes for a decent level

Made it to 20 range, might do magic right away aswell.
Heard that Vampire Slayer quest is good to do as it gives combat exp, probably will give it a try after i finish training magic.

I see that would be better to hit 25 so I can use Varrock teleport and after will go hunt that vampire B)

Can you guys tell me if I should try all skills to 40 after I get all combat skills to 40 too?
Cant figure out how to gain gold from game yet, just spending for supplies for the moment...

you wont make much low level but there is tons of guides to make bond type money in a few hours lootin certain items and yada yada. most of those things give no xp tho

Will check later, have 120k gp which could use to gain a little bit more hopefully.