Sometimes you are presented with the ability to use multiple draw and/or search (from deck) effects in the same turn. The order in which you use them can have an imperceptible influence on the final quality of your hand after all those effects have resolved. Here we'll go through some tips for improving your hand as much as possible.
This section is structured in the order that you would use those effects, so you can read it like a checklist: start from the top, see if the first item applies, and continue downward.
Example: Suppose your hand is 4 cards, including an Energy Removal 2 and a Professor Oak's Research. If you play Professor Oak's Research right away, you will have a new hand of 5 cards, 1 more than what you started with. If you use Energy Removal 2 first, your hand goes down to 3, so Oak will be a net +2 instead.
Some exceptions:
Don't play cards which do not affect the board (e.g. Pokémon Reversal when your opponent's only Benched Pokémon is a free retreater), or even affect your board negatively (e.g. Warp Point when your Bench-sitters are worse than your Active), unless you are very sure you will never need those cards for the rest of the game.
Don't play Pokémon if they are Pow fodder.
Don't play Stadiums if you need them to remove the opponent's Stadiums, but the opponent doesn't have a Stadium in play yet.
Don't play Energy if the situation is desperate and your only out is to spike a better Energy off the draw effect than the one that's currently in your hand. For example, you might currently have a Metal Energy in hand, but if that will not win the game, and only Double Rainbow Energy or Boost Energy would, don't play the Metal Energy.
If there are limited search targets in the deck (either there's literally only one, or there are several but you don't want multiples), then you do not want to draw into a search target and render the search effect worthless. Thus, you should search before drawing. This also thins the deck of dead draws for the draw effect.
Example: Suppose you have one Onix in play, ready to evolve, and your Trainers for the turn are Rocket's Poké Ball (to get Dark Steelix) and Mary's Request. Suppose the only targets for Rocket's Poké Ball are 4 Dark Steelixes, and you don't need more than 1 Dark Steelix right away because you only have one Onix. If you play Mary's Request first, you might end up drawing into 1 of the 4 Dark Steelixes naturally, and then Rocket's Poké Ball can only get a dead weight. If you play Rocket's Poké Ball first, Mary's Request has 3 (not 4) dead Dark Steelixes to draw into, and is more likely to draw into something useful, like a Cessation Crystal.
If you use another draw effect first, and the card that you draw can't be played immediately, that draw is wasted when you play the shuffle + draw effect. So, save the draw effect for after the shuffle + draw effect.
(If you play a search effect first, you should know what you're going to get, and be able to play the search target before the shuffle + draw effect.)
If you're using Milotic δ's Sharing without prior knowledge of your opponent's hand, a good rule of thumb is to assume that you're going to hit Rocket's Admin. with it. Every deck plays Admin, many at 4 copies, and your opponent would likely have spent other Supporters (like Basic/Evolution searchers) earlier in the game.
Magcargo's Smooth Over sets up your next draw, so you will want to follow it up with a draw effect. However, if the next effect is a shuffle + draw OR a search (which shuffles), then it will undo Smooth Over, so you want to reserve Smooth Over for after those have taken place.
Planning turns with Magcargo can get confusing if you have Magcargo, another Pokémon with a draw Poké-Power, and a Supporter. That's 3 effects to order. Save mental effort by batching Smooth Over with the draw Power and treating them as a single general search effect, so you only have to order 2 effects.
An exception is against Stantler, which can Push Away, and is likely to have Cessation Crystal equipped on the next turn. Smoothing Over for Windstorm and drawing it immediately leaves it vulnerable to Push Away, so you can Smooth Over last and leave a Windstorm on top of your deck instead. (Note that your opponent can still undo Smooth Over with Admin.)
Random draws off the top of your deck don't give you a choice as to what you get, but Wishing Star does. You'll want to use the effects that you cannot control first, followed by the effects that you have control over, because the former can end up providing valuable information for the latter.
Example: Suppose you just need one Energy, and your draw effects for the turn are Wishing Star and Mary's Request. If you use Wishing Star first and grab an Energy card, you risk drawing into redundant Energy when you use Mary's Request later. If you use Wishing Star first and don't grab an Energy card, you risk not drawing into any Energy at all with Mary's Request. However, if you use Mary's Request first, you can adjust what you get with Wishing Star: if you found Energy with Mary's Request, then you can ignore any Energy in the top 5 when you Wishing Star later, and if you didn't find Energy with Mary's Request, then you know that you must prioritize Energy when you use Wishing Star.
If you need two or more cards (which may or may not be duplicates) and have a draw effect plus a search effect that can get either one, then you want to try to luck into one of the necessary cards with a draw effect first. If you search before drawing, you thin the deck of hits, so avoid that.
Example: Suppose you need to get a Medicham ex (out of 4 remaining in deck) and an Energy (out of 15 remaining in deck, including Holon's Magneton). Your draw/search effects for the turn are Wishing Star and Professor Elm's Training Method. If you Elm for Medicham ex first, you have 15 hits remaining for Wishing Star. If you Elm for Holon's Magneton, that is even worse, as then you have only 4 hits for Wishing Star. However, if you Wishing Star first, you have 19 hits for it — you just need to hit Medicham ex OR Energy with Wishing Star, as Elm can get Holon's Magneton or Medicham ex, depending on what you're missing, to complete the set.
If you have search effects that can only get certain types of cards (e.g. Professor Elm's Training Method), and search effects that can get any card (e.g. Pidgeot), assuming perfect play, you'll know which cards you want to get with both, and it doesn't matter what order you use the searchers. However, if you don't know your deck that well, the safer play is to use the specific search first, then the general one. You might see something that changes your mind as you look through your deck on the first search. If it's something that the specific search can't get, you'll be glad you saved the general search for it.
Reminder: if search targets are limited, then specific search effects should be queued before draw effects.
General search effects are the most powerful — regular draws are random, Wishing Star doesn't see the entire deck, and specific search effects are restricted to certain targets. Thus, you will want to save them for last, after seeing the most cards and lucking into anything you could have lucked into with the other effects.
Reminder: if you have a Magcargo Smooth Over without a draw effect to follow it up with, then search first and Smooth Over last.