Lincoln's primary objectives would likely be:
Admission of Kansas into the Union as a free state. IOTL, this was done in January 1861, but only because the southern delegates opposing it had left. Likely, this will be accomplished, but at a later time and probably requiring some kind of concession to the South in another area.
Passage of the Homstead Act. This was passed in 1862 IOTL. Again, this was opposed by the South because slaveowners did not want yeoman farmers out west. They wanted land to be used in large plantations. They feared both losing their own whites that would settle out west, and new free states. Might be possible in the second half on his term if Lincoln can win Southern moderates - either Democrats like Andrew Johnson hostile to the slaveocracy or newly minted Southern Republicans from old Whigs in the Upper South who win in the 1862 Congressional elections.
Funding the Transcontinental Railroad. IOTL, this was accomplished by the Pacific Railroad Acts of 1862 and 1864. Most likely this can be achieved, but the southerners will need to be pacified by giving them a southern pacific railroad in order to win support for Lincoln's other agenda items.
Restricting slavery from any new territories. Of all the acts, this will be the hardest to accomplish in Congress. Most likely Lincoln will use a variety of measures to slowly restrict this until the Republicans can win sufficient control of Congress.
Outside of these legislative measures, Lincoln likely concentrates on establishing a Republican Party in the south through patronage jobs and wooing southern Whigs in the Border States and Upper South. Without secession, the Republicans can probably organize a local party and begin winning local elections in Missouri, Kentucky, Maryland, Delaware, and Virginia by 1862.
None of those states voted for Lincoln in the 1860 Presidential election. His votes were as follows:
MO - 17,028 or 10.3%
KY - 1.364 or 0.9%
MD - 2,294 or 2.5%
DE - 3,822 or 23.7%
VA - 1,887 or 1.1%
There is also the possibility of building some party in Tennessee, Arkansas, and Arkansas. No ballots were cast for Lincoln in this state, but probably a small party competitive in regional elections has a chance to grow. Perhaps local office in the mountainous areas with low levels of slavery, and perhaps even state representatives or even the odd Congressman, but probably not in 1862.
No real party will be able to grow in the Deep South, although we will see the beginning of votes there. But there is no chance of holding any elective office.
The southern Republicans will be on the party's moderate faction, but members will need to support its anti-slavery platform which will cause social upheaval in the south. Slowly, being against slavery will become socially acceptable.
Assuming the Republicans can build some kind of competitive party in the south, it can silence its critics that it is a "sectional" party. Lincoln needs the Republicans to win elections in the Border States and Upper South. I think he can do that in the Border States that voted in Constitutional Unionists. It'll be much tougher in Virginia and Tennessee, but there is potential in the Appalachian region.
The real question is what happens to the Democratic Party. Do they continue to have trouble at party unity? Most likely so, but that will make it easier for Lincoln to work in Congress, but shouldn't affect the local elections. Nevertheless, Lincoln needs a good result in the 1862 elections if his presidency to be a success.
The tenure of the time was against two term Presidents. Whether Lincoln wins depends on the Democrats splitting again. But he proved in 1860 that he can win the electoral votes with the North alone, and he might be able to pick up Missouri and Kentucky plus Kansas if it is admitted. That assumes Lincoln is kept as the nominee. More radical Republicans might want to replace him - another reason Lincoln needs to establish a southern Republican faction that is more moderate.