I'd like to add a few comments based on my experience using mainly dried yeasts over 40 years:
1) Inevitably you will strike the odd yeast sachet, that should be fine, but will not start.
I don't know why; it can happen regardless of brand or age, & the suggestions offered by others in this thread may decrease the chances of this occurring.
It's highly likely that if you follow the usual procedures, & simply pitch direct onto your aerated wort, within the acceptable temperature range, you will never see this problem again.
2) Personally when brewing from kit products, & when using 5 or 7 gram dry yeast sachets included with the kit or bought separately, I always use TWO sachets, & in my case I just pitch onto a well stirred wort at 25 to 27 deg C.
I can't recall either a slow start or start failure since I adopted this procedure about 15 years ago.
3) Despite the knowledgeable advice here, I would say after 36 hours of inaction your brew is likely "lost", in other words by the time you manage to start it some infection will be present & is likely to impact adversely on final taste. So what to do?
Well for the relatively small cost of a sachet of yeast, pitch again & let your brew complete. After the yeast has settled draw off a portion & taste.
If its good green beer you are lucky & you can expect a decent matured gassed beer. If you taste anything unusual or something you feel you would always notice when drinking, sorry dump the lot! If in doubt dump it!
4) If I am correct & at least for now you wish to continue using kits, you should carefully monitor the "time to see first bubbles".
You are aiming for 4 to 12 hours assuming you pitched dry at the correct temperature & assuming you used two yeast sachets.
If the time exceeds 12 hours, something is definitely wrong, & you need to investigate very carefully or your brewing career might suffer an unnecessary setback.
5) Some dry yeasts do start more quickly than others & some are also significantly more reliable. Generally they have an excellent life under refridgeration, I have successfully tested 4 years old sachets with success but would not seek to pitch yeast of that age without a newer sachet as backup - just common sense. Very recently I found some old sachets in a section of our fridge. I called the (kit) manufacturer who advised "don't worry they have an excellent life, but since you clearly support our product, we will happily send you some fresh stock". 4 days later a parcel of 12 sachets arrives!
6) I used to accept the advice of my local brew shop about which yeast sachets were the best. Now I realise they have to feed their family so they recommend whatever is best for them & they don't know anything about lower cost alternatives. I have tried liquid European yeast supplied on a nutrient slope (whatever that is) & many other yeast including bakers yeast. Sure you won't get the optimum taste that will allow entry into competitions. But for many just brewing reliably, & producing decent tasting, modest cost beers is sufficient.
Hope this comment helps & to more advanced brewers I hope you don't see this as dumbing down your science.
Rgds all Peter O