Having just read Grant's original memoirs--the original Mark Twain edition--and Jean Smith's splendid biography of Grant as well, I probably can add my own observations here.
Grant's strategy was always to cut off his enemy's supply lines, which in those days meant destroying the railroad lines supplying the enemy. Often with Sheriden's cavalry. He did this in Vicksburg (also using the Union Navy to control river routes and to transport troops), and finally in Richmond--cutting off Lee's forces around Richmond like a giant fishhook until Lee had no place to go--and no supplies.
Sherman's strategy in the South was similar--destroy all supplies and supply routes to the enemy--hence the scorched earth policy.
Grant was an old supply officer in the Mexican War, and he understood the importance of logistics--both supplying his own troops and cutting off supplies to his opposition.
His greatest generals--the guys he trusted the most--were Sherman and Sheriden. Hancock, too. Hancock, by the way, was a Democrat, and not of much help to Grant as President after the War. But a great General.
His best lifetime friend--his roommate at West Point--was General Longstreet--arguably the best Confederate general. Later, he commanded Union troops when Grant was President
to put down an uprising in New Orleans. Both Grant and Longstreet were members of the Literary club at West Point (Grant was President of the club). Grant spent more time in the West Point library reading the classics, than reading his textbooks. Hence, his excellent literary style in crafting his memoirs on his deathbed.
Regarding the politics of the time--one must be carfeul about labels--especially conservative, democrat, republican, etc. Indeed the politics were completely the opposite to what they are today.
In my view, Grant was as great an American leader as Lincoln. As President, he pushed the amendments to make blacks' citizens and give them voting rights. His main weakness was he trusted his friends and business associates too much--and some took advantage of that.