While Mike's answer is very well researched and written. I disagree with its challenges (though I don't disagree with his conclusions).
The biggest challenges that SpaceX will need to solve in my opinion are.
1. Re-entry heat shielding and Earth landings.
It is still working on mastering the landing flip without losing fuel flow. But it also has to show that orbital re-entry can be done safely and repeatedly without excessive heat stress degrading or destroying the Starship. Without both you don't have a fully re-usable launch system, and without full re-usability you can never have the cadence and low costs to support cheap interplanetary travel.
2. Refueling in space
It's never been done on the scale they need with the fuels they will be using. If they can't master refueling in space Starship can never take anyone beyond low earth orbit.
3. Long Term Life Support Systems
Crew Dragon shows they can make a safe life support system for space travel. But they have to show they can make a system that will last years without failing, and is easy to maintain and repair when millions of miles from earth.
These first three should be solvable problems, the question is whether they can be solved by 2025 and that seems very speculative. #1 and #2 have to be solved for the Human Landing System that SpaceX will be providing NASA to land on the moon. I expect them to master landings this year (2021), and start testing orbital re-entries by year end or in 2022. That gives them two years to master refueling for Lunar expeditions in 2024.
I think the real challenge to landing humans in 2025 is the need for prep/test missions.
4) Mars Prep Missions
Finding the right landing site isn't a real problem, there are a lot of great candidates. But SpaceX will need to send Robotic Starships to test landing on them, ensure that they aren't littered with boulders, or too uneven. But orbital mechanics (the relative position of Mars and Earth at any one time) mean we can only efficiently travel to Mars every two years. This means SpaceX needs to launch manned missions in late 2024 to get to Mars in 2025, which requires launching robotic missions in mid-late 2022. That seems like a very high order.
But beyond testing landing sites, the SpaceX plan is to land a good number of Cargo Starships packed with supplies and equipment before sending their first astronauts. The Cargo Starships would also require launching in 2022 to support a 2025 human landing, which would require combining test/cargo missions in same launches. That's super risky, if the first sites have bad characteristics and cause many of that first wave fail to land successfully there won't be enough supplies and equipment to safely send humans in 2025.
As for other challenges people may think of, here are the reasons I don't rate them as high a risk.
NASA studies have estimated that a two year round trip to Mars only produces a 4% higher lifetime risk of cancer. As long as Starship has a solar storm shelter for the crew to avoid the most dangerous radiation periods, risk should be minimal.
The risk of Regolith on the moon is that it's razor sharp dust because it's never had any environmental weathering. Thats not true of Mars, it's risk is Perchlorates in the soil and like radiation this is an overdone risk. It's not at a level that's directly poisonous, but if astronauts continually track perchlorate dust into their habitats eventually they'd suffer from thyroid issues. The simple solution is water, which drops the perchlorate into a safe solution. Spray the astronauts with before returning through the airlock, and spray the suits again before storing in sealed closets.
Mars is lousy with water at every latitude. If the tools they are sent with prove inadequate for digging up water or melting ice, they will certainly have tens if not hundreds of tons of distilled water in the Cached Cargo Starships to cover losses outside the life support systems.
7. Crew Training
Starships will be highly automated, the main training needs to be on the tools and equipment to be used on Mars. Apollo needed years of training because for the first time ever they were going to land on the surface of another world in a tiny lander with very little margin for error. Every potential risk had to be trained for and have a checklist for. Unlike Apollo, Starship missions will be equipped with a great deal of redundancies, supplies and equipment to deal with unforseen situations.
8. In Situ Fuel Generation
Again the cached Cargo Starships will have plenty of equipment and tools for producing methane and LOX for return trips. If they don't work, they are only two years away from another round of Cargo Starships with more supplies and updated tools and equipment designed directly to solve any problems that stymied the first attempts. Ultimately if the problem can't be solved, the solution is sending Starship Tankers to refuel on Mars itself. This plan will only be a problem if it causes SpaceX to struggle to recruit qualified crew for the missions, and I don't anticipate that being true.
I couldn't understand Buzz Aldrin's objections, other than Elon's cost estimates are targeted to decades away when launch cadence is at least hundreds per Synod, and its possible for Crew Starships to hold 100 passengers on Mars expeditions. They are also highly aspirational and not necessary to enable the first manned exploration missions.
Starship launches should cost roughly \$20M each when fully re-usable (based on the partially re-usable Falcon 9 costs), which would mean each Mars sent Starship costs \$160M for launches (assuming 7 tanker launches). A complete Falcon 9 costs less than \$60M to make, the similarly sized Cargo Starships (with 4 fewer engines) should be significantly cheaper to build using Stainless Steel construction. Crew Starships will be more expensive with their sophisticated life support systems, but SpaceX should only be sending a handful per Synod at first.
These first four challenges are solvable problems, the question is whether they can be solved by 2025 and that seems impossible. I expect SpaceX to master landings in 2021, and start testing orbital re-entries by year end or in 2022. That gives them two years to master refueling for Lunar expeditions in 2024. But the first expeditions can't be manned. So the real challenge to landing humans in 2025 is the need for Site Prep/Cargo Cache missions.
The net result is that landing humans in 2025 looks impossible today. I think it's far more feasible to land robotic Starships on Mars to site test in 2025, land cache Cargo Starships in 2027, and then land humans in 2029. And each of these dates to slip two years if anything goes wrong.