In order to upgrade your girls you will need to collect miscellaneous items that they want. Some of these can be bought from the Girls' Store for bucks but most have to be collected by fighting other players.
Fights can be organised by clicking on the "Fight!" button on the right-hand menu, or by clicking on the "Show me" button next to the girl's objective on their upgrade screen. You are then presented with eight players to choose from. Their attack strength is shown; their defense strength is hidden. The players you can choose from will have approximately the same stats as you. Initiate a fight by clicking on "Attack!", which will consume 1 stamina point.
The fight screen shows your gang on the left; the opposing gang on the right. In the middle is the item you are fighting over. If it is a "?" then it is a random item generated by the game (which may be nothing!). Above the item is another "Attack!" button which causes the fight to actually take place; underneath is a "Surrender" button. You also have the opportunity to buy boosters to affect the fight outcome for 10 gold apiece.
Once the "Attack!" button is pressed the game determines a winner for the fight. If you win the fight you will gain a random amount of bucks; +5 experience points; +5 mojo; and the item (note that "?" items may be revealed to be nothing, represented by a zero superimposed with a red cross). If you lose you gain +1 experience points.
If you win an item it is placed into an invisible/inaccessible inventory! This is very confusing for all players, particularly new ones. The item cannot be stolen by other players (unlike collectibles). The quantity you have already obtained will be shown to you when one of your girls requests that particular item. Items are classified as common, uncommon, rare or epic and their drop rate is affected accordingly.
(Note: when upgrading girls it is recommended that you do not "use" the item for a girl immediately and instead use them when you have collected everything you need to upgrade the girl. This is because if you add another girl to your gang, or upgrade an existing one, you may decide that you would rather use the items to upgrade that girl instead. Once an item has been given to a girl, it cannot be retrieved to give to another one.)
The general consensus of players is that the attack/defense and relative gang size make no difference to the outcome of the fight - it appears to be equivalent to a coin flip. It is therefore recommended not to spend gold on boosters. Additionally, since stamina has already been spent to reach the fight screen, surrendering makes no sense since that would be an instant loss. You should always follow-through with the attack because the coin-flip mechanics may mean that you win against clearly superior odds!
One of the more common questions players have is how to equip items on girls - you don't. Each girl automatically picks the best available item in each of the categories in the Girls' Store for their use. This is already factored into your gang's overall attack/defense stats so you do not need to worry about it. Again, note that your gang's stats have no apparent effect on the outcome of the fight! Similar considerations apply to the clothing your character wears: ignore the attack/defense stats and dress your character with whatever you want to wear - if that interests you.