In regards to the people of God, who was the first-fruit? Abraham. Like Abraham (the first-fruit) was made holy, so also the descendants (the lump) of Abraham are made holy. Abraham believed God, and it was credited to him as righteousness (Gen. 15:6; Rom. 4:3). Where did that righteousness come from? It was from the Son of God, the Lord Jesus Christ, sacrificed on the cross, who fulfilled God’s covenant with his blood (2 Cor. 5:21). So, the first-fruit was Abraham.
Who are Abraham’s descendants? As Rom. 4:11 puts it, “He is the father of all who believe,” be it Jew or Gentile. Like Abraham (the first-fruit), anyone who receives Christ’s righteousness by God’s grace through faith is a descendant of Abraham (part of the lump). This includes every believer, all God’s people, for all time, even those who lived before Christ’s sacrifice (Heb. 9:15 & 10:4). So, Abraham’s descendants are all true believers in God’s covenant promises fulfilled in Christ.
Then, what are the root and the branches? The root is Abraham, and likewise the patriarchs in the context of God’s covenant with them (a metaphor similarly used in various ancient Jewish texts – e.g. 1 Enoch, Jubilees, and also Philo). The branches are all of God’s people, rooted in God’s covenant fulfilled by Christ.
Paul elaborates on these branches in vv. 17-24 through the illustration of the olive tree. From this picture of a tree, Paul shows that God’s people are all those who receive their spiritual nourishment from the blessings of God’s covenant of grace—initially just the Jews, and later, with the Gentile Christians—both of which make up True Israel.
Let’s look vv. 17-24 again.
17 But if some of the branches were broken off, and you, although a wild olive shoot, were grafted in among the others and now share in the nourishing root of the olive tree, 18 do not be arrogant toward the branches. If you are, remember it is not you who support the root, but the root that supports you. 19 Then you will say, “Branches were broken off so that I might be grafted in.” 20 That is true. They were broken off because of their unbelief, but you stand fast through faith. So do not become proud, but fear. 21 For if God did not spare the natural branches, neither will he spare you. 22 Note then the kindness and the severity of God: severity toward those who have fallen, but God’s kindness to you, provided you continue in his kindness. Otherwise you too will be cut off. 23 And even they, if they do not continue in their unbelief, will be grafted in, for God has the power to graft them in again. 24 For if you were cut from what is by nature a wild olive tree, and grafted, contrary to nature, into a cultivated olive tree, how much more will these, the natural branches, be grafted back into their own olive tree.
Paul goes all out on his illustration of the olive tree. Why? Well, not only had the Jews been referred to as an olive tree in the OT (for example in Jeremiah 11), but how the olive tree was tended and pruned precisely illustrated the situation with true Israel.
Sir William Ramsey wrote an article concerning this in 1905. He wrote that the process Paul describes was still in use in Palestine.
“In exceptional circumstances…it is customary to reinvigorate an olive tree which is ceasing to bear fruit by grafting it with a shoot of the wild-olive, so that the sap of the tree ennobles this wild shoot and the tree now again begins to bear fruit.” 
In the case of the olive tree in Paul’s illustration, true Israel is the tree, with the patriarchs as its roots, rooted within God’s covenant. These roots nourish the whole tree with the promises and blessings of God which are part of the covenant. Thus, all the branches on the tree benefit from them.
The natural branches are the Jews naturally descended from the patriarchs. Those who rejected God’s covenant in Christ did not bear fruit, because they were spiritually dead. Due to their apostasy, God cut them off. As v. 20 puts it, they were broken off for their unbelief.
Then, in v. 17, God engrafted onto the tree the wild olive shoot (the Gentile Christians). This was due to God’s grace and their faith in Christ. As such, they have become part of true Israel, receiving all the promises and blessings of the patriarchs and the covenant. As Paul wrote in Gal. 3:29:“If you are Christ’s, then you are Abraham’s offspring, heirs according to promise.”