Do you check out/research products you see advertised to see if the ads are accurate? Do you check them out for complaints?

I do check the reviews on items for sale on the internet.  However, I usually start with the ones that gave it only one star - those are the complaints about which I'm most interested.  I also feel that those reviews are more likely to be honest ones.

My scepticism depends on what is being advertised, and how.

I trust most of the statements in ads not to be lies, knowing they obviously talk up the product or service, but what often matters is what the advertising does not say. They cannot lie but they can, and many do, fade or omit qualifying details.
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For example, yes, a statement like "... only £29.99 a  month..." will be valid, but in very small print you find that price is before tax, a starting fee, limited to the first 6 months, or is tied to something else. So is truthful but incomplete. For nearly £360 a year.
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Or how about this: "XXXX - It Just Got Better!", XXXX's genuine poster slogan some years ago.

Apart from its illiteracy, it may have been correct in its own way, if you could determine what that was. It is not misleading though, just meaningless, because it did not complete the comparison! 
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Here's another, from one of those curious little household-goods mail-order catalogues. It is not a verbatim but more or less there. It describes portable-'phone and lap-top wallets alleged to "...  protect you and your family from the harmful radiation emitted ..." by these instruments. 

The wallets have metal-foil linings that supposedly block the electromagnetic fields the ad means. Now, this is misleading for two reasons. Firstly, those electromagnetic fields natural to anything electrical we encounter in daily life are harmless (the ads are exploiting people who don't understand "radiation"). Secondly, the ads don't tell you that screening the 'phone may merely stop it receiving calls! You may as well save your money and simply switch the equipment off. Further, carrying a switched-on lap-top in a foil shield may lead to it being damaged by overheating.  
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Or the kind of thing said to been once common in music-equipment ads, "... our precious-metals alloy loudspeaker leads give a frequency response of less than 1dB loss up to 80kHz..."

That sort of rhubarb needlessly cost some music-lovers with poor technical knowledge, good money for expensive gadgets which even if obeying such claims in laboratory tests, had no practical value. (For a start, human hearing does not reach anywhere near that frequency.) It was like a watch I once owned: of its ..." 20 jewels..." only 2 were in use, as bearings. The other 18 were in there only to make the claim stated on the dial, true; as I learnt after one had fallen out of its setting and jammed the works.      

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On the other hand, I do occasionally buy from various tool and other specialist-goods companies; but I trust their advertising because they have a lot to lose by being deceitful. Most of their customers understand the products too well to be taken in by the dealers using partial price claims, meaningless slogans or the wallet- and hi-fi gadget- snake-oil tactics.   

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The UK's Advertising Standards Authority will fine companies for ads that are wilfully misleading or break other rules, maybe not enough to hurt the company financially, but potentially enough to damage the reputation and possibly share value.  
Its own publicity slogan says an advertisement has to be "legal, decent, honest and truthful" - I think I've quoted it accurately - and does invite reports of ones suspected of being not.

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One thing the ASA is very hot on, are false or misleading health claims for specific products. I think those are actually illegal; but I don't think it can act against generic,  non-product claims despite those exploiting ignorance.

One example are those made for "natural replacements" for "refined" sugar. The supposed replacements, including honey, are still basically plant-sugar no better or worse than the white crystals refined from cane or beet sap. Similarly with "Sea-Salt": same as rock-salt, i.e Sodium Chloride, but at fancy prices, in fancy packets careful not to state the mineral is somehow "different". This post was edited by Durdle at April 4, 2021 6:27 PM MDT