Aluminum cuts like butter, and pretty much anything will start cutting it. No need for carbide anything here. Standard HSS bits are fine. Sharp is the key, if you've abused any bits you had before then its going to be hard to get going. Go high speed, and "peck" the drill (slightly retract) action to break up the inevitable large birds nest of spinning aluminum that will develop.
However, I somewhat suspect your travel mug may actually be stainless steel... its rare to find drink ware made of aluminum (it would impart a pretty metallic taste, and would make for a very very cold or warm cup as its an amazing heat conductor). SS is very hard, but will also cut fine with a good bit- unlike aluminum, run it at _slow_ speed and be patient. The chips will be much smaller and won't generally form a birds nest.
Tooling pedantry:
Titanium isn't a material used for tooling. Its used as a coating (over both HSS and cardbide). For most big box stores the coatings are 80% cosmetic, so its not worth using it as a feature unless you're looking at specialty tooling. If you do buy coated, ZrN is best for aluminum, TiN (Titanium Nitride) is fine. TiAlN (Titanium Aluminum Nitride) should be avoided for aluminum, as it will be a mess - under heat the coating will just fuse to the aluminum you have (aluminum sticks to its self really well). Save yourself the money and buy uncoated