You have too many places for the short amount of time. You'll waste too much time changing locations. Using taxis, you'll want about 1/2 hour to check out, get to the station, find your train and board. Most stations have security checks and if the train is nearly full there may be a bit of a line. Trains close their doors 2 minutes before departure. On arrival you may have a wait for a taxi and you may get into heavy city traffic esp. in Barcelona. Then you have to drop your luggage and check in before you can go out sightseeing. As a rule of thumb, add about 2 hours to the train travel time. 2 nights in a city gives you just one full day there.
Looks like your only good option to Barcelona is the 8.15 train arriving at 2.25 pm. That doesn't leave you much time on Sunday, so you only have 2 full days there.
Toledo to Seville. Looking at the schedule, your earliest options are:
6.50 Toledo, arrives Atocha 7.23. The next train to Seville is at 8.00, arrives 10.32
7.55 Toledo, arrives Atocha 8.28. The next train to Seville is at 9.05, arrives 11.26
Even without Segovia, you'll only have part of the 29th (jetlagged?) in Toledo and one full day. Then you'll have 1.5 days in Seville, 1.5 days in Granada and 2 days and a bit in Barcelona. And you'll spend a lot of money on trains. Use the Renfe site to check schedules and prices.
Looking at your posting history, you had planned a trip to Italy in April 2019 at a similar pace. How did that go? You also planned a trip for Oct. 3 this year - I'm guessing you scrapped that since you're obviously not spending all your time out sightseeing right now :-) Most of us find such whirlwind trips both tiring and unsatisfying, but that style does suit some people.
Even so, with only 9 full days on the ground and long distances between destinations, I'd recommend paring it down to 3 locations. From the airport you need to get to the Madrid Atocha train station and that's after passport control - hopefully you're only taking cabin baggage so you don't have big cases to check, wait for at the airport, and lug on the trains. AT the airport, you'll need to buy train tickets and then take a train to Atocha. Even if you opt for a (possibly) faster taxi ride (traffic can be a bear), you should probably still buy your tickets onward to Toledo at the Renfe office at the airport. It is not advisable to buy tickets at Atocha, the lines are notoriously long at the ticket offices and the machines are not always easy to use.