RX pin --> TX of FTDI
TX pin --> RX of FTDI
This should be the case for both programming mode and normal operating mode - always.
Other than that, check that your 3.3V supply can supply sufficient current.
Check that your FTDI (or similar) device works by sending characters in a serial terminal, and then just connect the TX back to the RX with a wire, and confirm that you now get an "echo" from the serial port.
Generally, I would not recommend the ESP-01 as a worthwhile beginner's device.
It's hard to set up for beginners, you have to provide the serial comms and 3.3V supply, you can't use it on a breadboard, and you only get a couple of GPIO pins.
An ESP-12 is about the same price, or a Wemos D1 board, or a NodeMCU Amica board, are all much easier to use.
Something like an Amica is slightly more expensive but gives you the 3.3V and serial already sorted easily for you.
GPIO15 is already tied to ground on the ESP-01. There are already internal pull-ups on GPIO0 and GPIO2, but you can try adding some yourself, like 10k or 4k7, and see if there's a difference.
Maybe try adding a decoupling capacitor on the 3.3V supply too.