The 1890's are an interesting time of transition for the US Army. In terms of promotion, the last of the Civil War generation of officers was finally retiring and the frontier was closed as the Indian Wars came to an end. The military was paired down to it's minimums, but at least things were leveled off a bit now and the glut of senior officers holding up the promotion pyramid were gone. Remember a lot of career Army officers in the Civil War who decided to stick around after the war had to to do so under the condition that they would accept a demotion. Congress sets the number of officers that are allowed at each rank, and a demobilized Civil War army full of Generals and Colonels who no longer had anything to command were a problem. For example, General George Armstrong Custer was a dashing and famous 1-star Brigadier General during the Civil War and was even present at the surrender at Appomattox. When he met his fate 11 years later however at the battle of Little Big Horn, he was a lowly Lieutenant Colonel, or 2 levels lower than his rank during the Civil War.
Now, things would heat up and expand again when the Spanish-American War would start up in 1898, but let's assume it's, say 1895, and that hasn't happened yet. The most junior officer in the military is a 2nd Lieutenant in the Army or Marine Corps, and an Ensign in the Navy or Coast Guard.
In 1895, the starting point for a new 2LT was about the same as it is now. You would be commissioned at or around 21 years of age, usually out of West Point. It could then take 3 to 5 years to get promoted to First Lieutenant depending on where you were stationed, how you performed and how visible you were. Out of sight, out of mind tended to apply when promotion boards came around. You could stay a 1LT for another 2 to 5 years and then get moved up to Captain. Captain was, and still is, the rank you are the longest as a career officer. For most officers in the late 1800's, Captain was their terminal rank and is the level you would retire from in a peacetime career. However, if you were hard working, talented and got noticed, you could move up to Major in about an additional 6 years, which is about what it still is today. You could then be terminal as a Major (still a respectable rank to retire from today), but you could then advance to Lieutenant Colonel 4 years after that possibly. If you got command of a Regiment, you would be moved up to full Colonel. After that, who you knew got you promoted to General.
Putting this all together, if you had started as a 2LT in 1895, you probably would have emerged as a Captain if you had smelled gunsmoke during the Spanish-American War. If not, then a 1LT by 1898. By the new century, 1900, odds are you would have made Captain no matter what. And there you would have stayed probably until 1914 when WWI started. Once the US made the decision to start building up to enter the war, all those long suffering career officers would have been moved along quickly with some long overdue promotions, as a cadre of core professionals were sorely needed to expand the modern force. By the time the US entered the war in 1917 you would have been a major, and you would have emerged from the war probably as a Lieutenant Colonel. There you probably would have retired from, unless you stuck around to the 1930's, where you would have made full Colonel, but an aging one..
All of the above is for an "average" officer during this time period, and of course there were many careers that were far better, and far worse, than this.