A mystery transformer!
Firstly, I see it is a leak transformer as the core extends in between the the primary and secondary windings.
I will tell you the procedure I go through with ANY mystery transformer straight out of the mystery transformer box.
First process is to determine how many discrete and separate windings it has. So let's say there are three wires hanging from one coil and four wires hanging from the other coil.
The three wire coil can be one of three things:-
1. A single winding with one tap.
2. A single winding with an electrostatic shield.
3. Two separate windings with two wires each and one wire has been ripped out by somebody being careless.
The four wire winding would generally be one of two things:-
1. Two separate windings with two wires each, not electrically connected.
2: One single contigenous winding with two taps.
Use your multimeter, (on the low resistance range) to determine which arrangement it is, as single windings will measure relatively low resistances to all connected wires and taps, all readings should be <1K or so. Electrically separate windings will measure open circuit to other isolated windings or electrostatic shields. (Always bear in mind there might be open circuit windings or shorted turns in an old transformer.) Group the wires or connections of each identified winding and its taps together, (if thay are wires hanging out, put tape around them with some I.D., Like, "Winding 1" written on it. If the transformer has solder lugs or screw terminals, use coloured felt tip pens to identify which windings are which.)
Once you have identified which pairs (or more) of wires belong to which windings it is time to determine the "voltages" of those windings. To do this requires LIVE testing, so take due care!
Voltages, generally the mains, needs to be applied to the windings...BUT there MUST be an incandescent lamp in series with the maims on the active side. This process is easier if you own a Variac, but if you don't, some useful information can still be gleaned.
IF YOU DON'T HAVE A VARIAC
Place an incandescent lamp with a wattage rating somewhere between the transformer's VA rating, (roughly determined by the size of its core and comparison with other transformers of known definite VA ratings) and about half the estimated VA rating. Also, make sure the lamp's voltage rating is equal to the mains voltage at your location!
So if you estimate it is 200VA, use a lamp in the 100-200w range.
Find one of the windings you identified above with the multimeter, try to determine which wires are the very ends of the winding by measuring for the maximum resistance combination...then apply the mains to this winding VIA THE INCANDESCENT LAMP IN SERIES!
If the lamp does not light at all, or the filament just glows a dull red, then the voltage of the winding you have chosen will be either greater than the mains voltage, (no glow) or about the same as the mains voltage, (dull glow). If the lamp lights strongly, by more than one third brightness...the winding has a voltage LESS THAN the mains and if this winding is connected directly to the mains, either the breakers will trip or the transformer will buzz fiercely, get hot and possibly catch on fire.
Go around all your windings like this until you locate one, or a combination of taps on one that gives the very dim glow on the series lamp. Once you find this winding, good chance you have found the primary.
Turn it off, set your multimeter to the high AC volts range and croc-clip it across one of the other windings....if the other winding has very thin winding wire (if you can see it, which is not always the case), much thinner than the primary, it may well be a high voltage winding...if you think it will exceed the maximum range of your multimeter DO NOT proceed as it will permanently damage your multimeter. If you suspect a very high voltage secondary...using well insulated long nose pliers or plastic chopsticks....still with the lamp in series with the primary...turn it on and use the chopsticks to short out the ends of the suspected HV winding....the lamp should light up nearly full bright when the ends touch, when you separate them if there is a firey arc...then it is a HV winding....of there a crackly sparks when you touch the other winding's ends, then it is probably a low voltage <50v.
If you feel confident the highest voltage winding on it is less than your multimeter's maximum rating on the AC range, then feel free to measure the secondary voltages and their taps...and you will then get all the secondary voltages directly.
With leak transformers, such as depicted above, shorting the secondary side will not fully reflect the short back to the primary side and the series lamp will not come to full brightness. In this case you can place your multimeter on the AC current range and measure the short-circuit secondary current....which if it is intended to drive a discharge lamp, will be somewhat equal to the lamp's starting current and about half the lamp's running current.
IF YOU DO OWN A VARIAC
Things are somewhat easier. Place the Variac "upwind" of your test set up, i.e. plug the Variac into the mains, plug your test set up of lamp and unknown transformer into the Variac's adjustable output side.
In this case, turn the Variac down to zero, connect to any known contiguous winding on the unknown transformer..then turn up the voltage on the Variac slowly and watch the lamp carefully...if you get to the maximum setting and there is no glow, use insulated handle long-nose pliers or plastic (no wooden) chopsticks to pull one wire off the transformer. If there is no spark when you break the circuit the winding you have chosen is either open circuit or you have chosen the ends of different windings...if there IS a spark, it should be a firey one, this means the voltage rating of the winding chosen is higher than the maximum setting on the Variac.
If the lamp does light up, the voltage setting on the Variac at which the lamp just glows is the same as the voltage of the winding you have connected to.
If there are shorted turns inside, the lamp in series will ALWAYS glow regardless of which winding you are connected to with the supply voltage applied equal to 29% of the mains or more. In this case, chuck it to scrap metal or rewind it.
GOOD LUCK, I can't tell you anymore than this!