Welcome to GFE,
davidhsamson3.
Good to see you've been perusing the Boards for help in planning your trip. No doubt you've read some recent Posts about pre-bookig accommodation, and this still applies to September as it's not quite the off-season. In fact, the "season" is getting later in Europe and extending all the way to the end of October. There are quite a few Autumn/Fall festivals to be had in Europe, and these are popular with locals and visitors, so I'd suggest that you book your accommodation ahead of time if you want to stay in specific hostels and stick to your budget. This will apply particularly to the first part of your trip as it's still busy then.
As it will be getting cooler in the north of Europe, you would be wise to reschedule your itinerary to visit all of those places first (eg move Italy, Spain and Portugal to the end). Is there a reason you have cities in the order you do? (ie events, friends to visit?). And a point about Greece - Athens is "open" all year round, but the islands close down in September and it gets very windy and cool, so if you were planning a beach trip to Santorin, you'd be disappointed at that time of year. Perhaps leave it out for the next trip and include a few more islands (Mykonos, Naxos, Paros) at the same time.
If you haven't booked your ticket already, look at an open-jaw from Dallas to AMS, then out from, say Madrid back to Dallas. If the cost is still within your range, it would save backtracking and let you make the most of your time.
To address your list of destinations:
Netherlands- Amsterdam for 2 full days, unless you want to do day trips (Den Haag, Groningen..)
Belgium - Brugge/Bruges is a good choice, also spend a day or at least a lunch in Brussels, plus if you base yourself in Brugge you could do a day trip to the WWI battlefields if that interests you.
France - Paris for 4-5 full days, again depending on your interests. Nice at that time of year won't be super-warm, but you could still sneak time in on a pebbly beach if that's what you want. Otherwise, look at smaller villages like Beaune or even Avignon if you want somewhere a little different.
Spain -
herrbert is our resident expert here, however to get things started I'd suggest Barcelona for 3 days and Madrid for 4 days (this will allow for day trips). Madrid isn't near a beach, in fact it's a very long walk
so if you're wanting beaches, and again the Autumn weather can be fickle (and if you move Spain to the end of your trip then it won't be beach weather at all), look at Costa Blanca (south of Valencia) or even along the Andalucian coast as it will still be warmish (the Costa del Sol is on the southern end of Spain and is super touristy, but super beachy, so it's a trade off). A lot of people like the Costa Almeria as it's not too developed, particularly the eastern part. As far as "less tourist spots"...hard when that's why people go there!
Portugal - again, there are others who have more information than me on this country, but Lisbon and Lagos are good choices, both of which would need 2 full days.
Switzerland - if you're looking for adventure sports, then Interlaken is your spot. You could also stay further up the mountains in Lauterbrunnen or Gimmelwald as they have stunning scenery and also offer paraglinding, mountain biking, etc.
Italy - skip Milan and Genova and instead of Portofino, look at one of the villages in Cinque Terre. There's nothing wrong with Portofino, but the CT are pretty and offer a range of different villages to stay in. Of course, it's your choice, but don't overlook CT (plus you can visit Genoa for a few hours from here) as it's not as busy in Autumn. Just watch the storms at that time of year, particularly in the afternoon/evening. Rome will need 4 full days (5 if you plan a long day trip to Pompei) and Venice 1-2 full days. You haven't listed Florence, which is worth a visit if you want to see the museums and all things Renaissance, otherwise stay in Siena for 2 days as you'll have a smaller town will a country feel. There are lots of smaller places in Italy to visit, but very few are "off the beaten path" anymore. Lake Como, Lake Garda, towns in Tuscany and Umbria, plus towns in Puglia.
Sicily- (which is still part of Italy!) IMHO needs 10 days to explore the island. But, depending on what you want to see, you could spend 5 days on the island and stick mainly to the east coast (Catania, Siracusa, Taormina).
Greece - see above on the islands. Athens for 2-3 days (depending on how much sightseeing you plan) would be fine, plus time in the Peleponnese (Tripoli) would be good for a few days, again depending on what you want to see. You'll cross the Corinth Canal on your way from Greece, so include that in your list.
Czech Republic - Prague for 3 full days is a good start, plus you could always do a trip out to Kutna Hora for a day if you're interested.
Sweden - what places were you considering?
Denmark - as above?
Scotland - Aberdeen doesn't have much going on (I know, my brother used to live there!), and same for Perth. But the surrounding areas are interesting enough, plus if you're tracing family history then they may be applicable. Glasgow is good for a day or two (depending on your interests) and you should consider Edinburgh (you could visit Stirling and even Falkirk Wheel on your way from Edinburgh to the east coast or Highlands, depending on your decision).
Ireland - why not all three cities? You could travel to Belfast, on to Galway, then back to Dublin and then fly to your next destination from there. Belfast for one full day (unless you're visiting the Giant's Causeway), Galway for 2 if you want to spend time in the Connemara and Dublin for 2 full days.
UK - Manchester for 1-2 days (depending on your interests or reason for visiting - football? family?) and London for 3-4 days (again, depending on your interests, but that's a good amount of time for LDN). Are you considering anywhere else in England? I ask because depending on how your reschedule the order of your itinerary, you could visit places like York or the Lakes District on your way to/from Scotland.
You haven't listed Germany at all - is there a reason why? Look at Berlin and Munich as a minimum. Even though they are larger cities, they definitely warrant visiting, just be aware of Oktoberfest in Munich in the last 2 weeks of September and 1st week of October - accommodation is already booking out.
Once you've looked at the suggested cities, and the other Members have given there suggestions, you will have an idea of what is possible in the time you have to travel. Also look at a mix of flights, trains and busses for your travel, depending on the route you choose.
Happy planning!