The London Northwestern Railway are twice an hour, M-F, different times on Saturdays and fewer on Sundays, from Euston to Birmingham New Street. They are slower because the Avanti West Coast trains are tilting Pendolinos which travel at 125 mph and only stops briefly at a few intermediate stops (Watford Junction or Milton Keynes Central or Rugby) and Coventry, Birmingham International, and Birmingham New Street, whereas the London Northwestern commuter trains are no faster than 110 mph, they go via the secondary line through Northampton, and they have several minutes' dwell time at both Milton Keynes Central and Northampton, and several more stops than the Avanti West Coast trains which are the very long distance express trains.
The xx:15 LNWR trains off Euston are slightly slower and require a cross platform (usually) change at Northampton. The xx:49 trains go straight through. They don't offer reservations and while not so fancy on board do tend to be quite a lot cheaper.
Both Avanti West Coast (which goes 3 times an hour to Birmingham) and London Northwestern are fully electrified and are very quiet and comfortable. The LNWR windows line up with the seats and are much larger than those on the Pendolinos of Avanti (formerly Virgin) which don't always line up well.
The Chiltern trains use the very small and quite beautiful Marylebone station in London and call at both Birmingham Moor Street and Birmingham Snow Hill stations in Birmingham, call at several intermediate stations on the way, different ones at different times of day. The trains are diesel, and most have only one class of travel, better than the usual level of Standard but quite a way below First.
The cheapest tickets on all three are called "Advance" and they must be bought in advance, the further ahead the cheaper, although many cheap ones on LNWR can be bought quite close to travel if you are lucky, and are non-refundable. They can only be changed before travel (for a significant fee) - never on the train - and are specific for a specific train at a specific time. Miss that and they are a nice bookmark. Off-Peak singles and returns are next cheapest, and Anytime, which can be used at the rush hour, are the most expensive. Anytime tickets on Avanti West Coast are eye-wateringly expensive, but they can be used all day and refunded.
It all boils down to your circumstances, what part of Birmingham you want to get to, and if time or money are more important to you.
If you are going anywhere beyond the centre of Birmingham there is an extensive network of local trains operated by LNWR's sister company West Midlands Trains. There are also trams and buses.
Feel free to ask away, Alice, I worked on all those trains for decades and know the route and the local Birmingham routes like the back of my hand. Happy to help.