your options are....
1. TIG set. Needs (expensive) gas, needs skill, needs clean parts to weld.
2. Cheap ( AC transformer type) stick welder. Needs sticks, needs skill to make a good job. IME single phase transformer sets struggle with 4mm electrodes, which is about what you should be thinking of using with 6mm steel.
3. Posher (inverter based) 13A stick welder. Will handle 4mm electrodes, is more portable, is more likely to make good welds (it is a better regulated supply, with DC available in most cases)
4. MIG set. Cheap, easy to use (much easier than stick welding), needs to be fed gas and wire.
My personal choice for general workshop use (13-16A single phase supply) would be a MIG set with about 150A rating. If you use straight CO2 gas (OK for steel) you will need one ~7kg bottle every six months or a year, even if you use the welder several times a week. Don't get sucked into a BOC contract, there are alternatives; I get my gas from the local car spares store; it is about £30 a fill (for ~5-7kg of gas) once you have bought the bottle (similar cost) and there is no rental charge. BTW a comparable size argon bottle typically holds half or a third of the gas by weight; CO2 is liquid in the bottle, Argon isn't.
Second favourite would be an inverter type stick welder. I have a portable kit that all stashes in a small suitcase, and the power supply is a box about 3" x 7" x 8" in size. It welds beautifully, but (as with all stick welders) the heat input to the welds is a lot more than when using a MIG welder. It will manage 200A current from a 13A socket. You will need ~two passes to weld 6mm steel full thickness.
If I make the same thing by stick or MIG, the MIG welded part is better and distorted less. You can weld 6mm steel with a 150A (single phase) MIG set but it requires careful setup and two or three passes. Whilst higher powered MIG sets will weld 6mm steel in one pass, these (or indeed any electric welder that would easily manage 6mm in one pass) won't run from a 13A socket (or a 16A one come to that).
FYI if you are in the range of 250A or more, MIG welding can operate in a 'spray transfer' mode that is very different from the mode employed in lower power sets. Spray transfer MIG is the most commonly used arc welding method in the manufacture of all kinds of mass produced brackets and steelwork in thicknesses of ~3mm to ~10mm.
BTW you may be thinking of gasless MIG. This is the runt of the litter; it makes poorer quality welds, is easily the most expensive in terms of consumables (the wire is x5 more expensive, and often only yields ~50% by weight of deposit...) and it is messier and more toxic than standard MIG. It would be my least favourite choice for workshop use.
Also NB for occasional use you can probably accept a welding set with a low-ish duty cycle; no point in buying more duty cycle if you don't really need it.
cheers