How to solder connections to this board

Those are test-points, in your example gold plated ones, and are not intended to be used to solder wires onto them.

Normally they are used with a bed-of-nails using spring loaded pogo-pins to make contact to the circuit using testing equipment. Pogo-pins come with a variety of tips for contacting with various features on the board.

Note the use of the large pointed pins in the above image that go through the mounting holes in the board that act to align the board with the pins. The big fat ones are keyed to the edge of the board to get it aligned close to the smaller ones that go through the holes.

The alignment pins are longer than the pogo-pins so the board is in the correct position before they make contact.

Sometimes double sided, "suitcase", fixtures are used...

While it's true that these would be contacted by pogo pins rather soldered in production, for a personally owned unit (or indeed, for the original firmware developers') soldering would be quite reasonable and likely.

Fine gauge silicone insulated wire is best for this, since you avoid the whole complication of potentially melting the insulation while soldering with only a couple of millimeters of stripped length, and you can with a little care put a crimp connector on the other end to connect to your programmer, USB/serial adapter, logic analyzer, or whatever. But you can also use wire-wrap wire with a bit more care. Others prefer magnet-type wire insulated with a solder-through insulation.

While the pads are close together, compared to what they could be, they are really not that tight at all - be glad you aren't trying to pick up the signals from the 0402 resistors or the pads of that little QFN chip.

Pre-tin the wire (I tend to start overly long and trim to about a millimeter exposed after tinning), possibly pre-tin the pad, and you just need to hold the wire in place while you touch it for a second with the iron.

those are gold plated test points

you would not normally solder wires to those

but if you really, really, really have to, apply solder to the pad, then apply solder to end of wire, then join the two