3 week Europe trip

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Hi there, my husband and I are planning our Europe trip for August this year and I was hoping for some help on the best way to book our tickets and which would be the cheapest way to do it... We fly into London but are going straight to Paris, we will be using Eurostar as our hotel is right near Gare du Nord, after this it follows:

Paris -> Zurich -> Munich -> Vienna -> Budapest
then flying from Budapest to Manchester on a cheap flight to stay with family for a few days and then make our way to London for a couple of days and then fly back home. We are staying for 3 days each in Paris and Zurich and then 2 days in the others. We hope to travel in the evening to our next destinations to make the most of the daylight.

Suitable Eurail pass seems to be a 5 country Select pass (5 days travel) at $AUD481 (~375euro) each, we would plan to use the extra day for a day trip to Lucerne. I can't seem to look at individual tickets yet for our specific dates to see if it will be cheaper to buy these instead... My main concern for individual tickets would be locking into a specific time and also how beneficial is it to travel first class with the eurail pass over second class with normal tickets? Any other tips in general appreciated

Ok, so let's go step by step:

London-Paris: in case of traveling with EuroStar Eurail pass is pretty much useless, since you need to pay additional supplement which in many cases higher than one-way ticket. So it's just makes sense to visit EuroStar website and book some time in advance plain one-way ticket (can be as low as 39 GBP) or same thing, but via French website (tgv-europe.com), from 42,5 euro.
Paris-Zurich: same idea here, on tgv-europe website you can catch tickets for August for as low as 41 euro per person (that is as of now), but time to time SNCF (French Railways) have additional promo campaigns etc. and tickets start from even lower prices. Alternatively buy such ticket on SBB (Swiss Railways) website, their you can get it even for 31 CHF (app.25 euro)
Zurich-Munich: and again same thing with this route. Purchase in advance allows you to catch discounted tickets, either via quoted above SBB website (from 48 CHF) or DB (German Rail) - from 39 euro. Though here full fare isn't that bad - 62 CHF
Munich-Vienna: DB website and discounted ticket that start from 29 euro per person
Vienna-Budapest: here discounted tickets start from even lower price - only 19 euro. Can purchase it via Austrian Railways (OeBB) international tickets eShop. We even have a guide on how to manage it - http://europetrainsguide.com/Advice/How-to-Buy-Train-Tickets/Austria-OeBB.html

As you can see, if you rely just on discounted point-to-point tickets you can spend as low as 150 euro per person for the whole itinerary or maybe a bit more (in cases when you won't be able to catch cheapest promo tickets). Obviously buying such tickets means tie yourself to a specific date and train, but that is the price you pay for cheap train travel. As result you just need to make a choice - more flexibility, but higher price (-->rail pass) or almost no flexibility, but lower total price.
Also remember that besides having Eurail pass for leg London-Paris additional price for you will be 89 euro (pass holders fare for 2nd class), while for Paris-Zurich you'll require additional seat reservation is another 9 euro.

We hope to travel in the evening to our next destinations to make the most of the daylight.

Well, though trains in Western Europe nowadays do travel fast, but it's still not always about an hour or two, so for instance Paris-Zurich is at least 4 hours, same for Zurich-Munich and Munich-Vienna. Also consider that usually evening connections are the most popular, so it's not just about crowded train, but promo tickets for evening connection do sell out faster and chances to get them usually lower

...also how beneficial is it to travel first class with the eurail pass over second class with normal tickets?

We usually specialize on a budget travel options, so it's morally difficult to recommend 1st class to anybody   But anyway, in case of Western Europe main difference is usually just about 2+1 seats (instead of 2+2) and some additional services (like paper or meal, like in case of EuroStar), while in terms of comfort you hardly will feel any huge difference.

...we would plan to use the extra day for a day trip to Lucerne.

For trip to Luzern you can get extremely cheap SBB SuperSaver tickets for as low as 9,2 CHF one-way, so having a pass shouldn't be an obligatory thing in order to make a day trip to Luzern