I am on kind of a tight budget. I'm level 44 and am trying to get to level 55 at least. My armour includes a zombie shirt, steel square shield, and a steel med helmet. my weapons are a blurite sword and a mithril mace. I would like to know which weapon on which creatures would be the fastest way to level up.

Don't level up with those items. It'll take an awfully long time.

 

I suggest you make enough for Full Addy plus a weapon. Use one of your skils to make the money such as: Mining, Fishing or Woodcutting.

at that level i would get some better equiptment, and go train at hillgiants near varrock, those are what got me from lvl 50-lvl 73(right now)

dont train combat until you have money for at least a full set of the same armor. wearing random stuff isnt going to get you anywhere. go make some cash by WORKING. then buy a full set of mithril. use that mithril till you can use rune and WORK until you can afford rune. always use a simmy to train.

Thanks for the tips.

I suggest mining iron for some quick and easy money, don't sell it for below 100gp, go to world 1 in falador where you can sell it for 100ea. Get about 300 iron ore and you'll have enough to buy full addy :)