> I have 2 pewter tankards dated 1877. They are quite seriously tarnished.
Pewter isn't generally considered to "tarnish" (as silver does), but rather to "patinate". The difference is like garden weeds: a weed is just a flower in the wrong place. Patina on pewter is an even grey that is generally considered to be an inherent part of pewter's appearance. It would be remarkable to try to change this at all, once formed.
Pewter doesn't usually tarnish, unless it has some sort of spill upon it. Jugs used for flowers are the usual culprits.
Pewter may also be dirty. It might certainly want cleaning.
It's also possible that they're not pewter, but silver (or plate) and this silver has become so badly tarnished as to look like pewter! It's not common, but it's a great find when you do get one. So check the maker's marks, touchmarks or even a hallmark.
> Should I clean them? If so what with?
I would be amazed if removing the patina was appropriate.
If you care about these pieces, first acquire some charity shop teapots and practise. If you're lucky, you might even turn up a Tudric or something that's worth selling afterwards. BTW - charity shop tankards are usually modern, a different alloy and don't handle in a very similar way.
Cleaning can be done with a range of powerful chemistry, but just avoid mechanical abrasion. Pewter is soft, so even rubbing with a coarse cloth will affect that patina on edges or fine detail. If the pewter has any sort of surface texture, be very careful with it. You should also try to identify what the staining is, and choose your weapons accordingly.
If it's organic (i.e. oily) grime, then start with organic solvents. The old favourites of isopropanol, acetone, liquid lighter fuel, sticky label remover, brake cleaner and carburettor cleaner aerosols are all useful (in much that order). Work with cotton wool makeup pads. Don't use a cloth, because cloths get pulled tight and tight abrasives round over corners and destroy detail.
If it's cheap stuff in bulk, then just dishwash them! Finish (not Persil) does a decent job. Watch out for wooden, ivory or early plastic knobs and handles though.
Ultrasonic cleaning is a good technique for brooches, badges and small stuff small enough to submerge in your cleaner. Don't semi-immerse big pieces or you'll get tidemarks.
Limescale-like staining can be shifted with a range of chemicals, from steradent tablets, through perborate laundry bleaches (i.e. "oxygen" bleaches as white powders, rather than chlorine or hypochlorite bleaches), up to sulphamic acid based kettle descaler. You can also have a dental hygienist take a go at it, because their Air-flow machines (miniature bicarbonate of soda sandblasters) are great for this.
If all else fails, there's oxalic acid, aka Barkeeper's Fiend. Dead handy, but it'll trash a lot of the patina. For use when it's bright green with copper verdigris from something else and the alternative is a mechanical repolishing.
Strong alkaline cleaners, like the old Mr Muscle (no longer available) used to have their place here too, although don't say I didn't warn you to test and practice beforehand. Alkalis are a good way to attack particularly recalcitrant greasy stuff.
This is about the point when you transition from cleaning to repolishing. Repolishing is always bad in antiques, because it removes material that can't be replaced. The difference between a museum and an antique shop is that a museum will go to insane lengths to avoid this, but an antique shop wants a one-off restoration for "Shiny now!" and to just get it out of the door. Afterwards is not their problem.
Repolishing pewter is dead easy, as it's an easy metal to work with. Repolish with the softer end of the usual jeweller's abrasives, finishing up with tripoli powder. You might even use heavyweight abrasives like fine Garryflex blocks or 3M / Webrax scourer pads, but that's pretty heavy-handed and won't do inscriptions any favours. Modernist pewter might have scratch-brushed or matted surfaces, for which you use a nickel wire brush (use the right brush!), but not this sort of piece.
By this point, the surface is also "active" and is very sensitive to staining, before a future patina develops. Only handle it with cotton or vinyl gloves.
If you want to preserve the "as cleaned" look as much as possible, then wax polish it afterwards with Renaissance Wax (_not_ any other wax polishes) and don't handle it.